Decomposition of Cartesian Product of Complete Graphs into Sunlet Graphs of Order Eight

TL;DR

This paper establishes the precise conditions under which the Cartesian product of complete graphs can be decomposed into sunlet graphs of order eight, contributing to graph decomposition theory.

Contribution

It provides necessary and sufficient conditions for decomposing Cartesian products of complete graphs into sunlet graphs of order eight.

Findings

Derived conditions for decomposition existence

Characterized the structure of such decompositions

Extended understanding of graph decomposition patterns

Abstract

we define the sunlet graph is the graph obtained by taking one copy of the cycle and joined every vertex of the cycle to an exactly one pendant vertex such that the degree of each vertex in the cycle is three. In this paper, we establish necessary and sufficient conditions for the existence of decomposition of the Cartesian product of complete graphs into sunlet graph of order eight.

Full text

2010 Mathematics Subject Classification: 05C51

Keywords: Graph decomposition, Cartesian Product, Corona graph, Sunlet graph

1 Introduction

All graphs considered here are finite, simple and undirected. A cycle of length $k$ is called $k$-cycle and it is denoted by $C_{k}$. $K_{m}$ denotes the complete graph on $m$ vertices and $K_{m,n}$ denotes the complete bipartite graph with $m$ and $n$ vertices in the parts. We denote the complete $m$-partite graph with $n_{1},n_{2},\dots,n_{m}$ vertices in the parts by $K_{n_{1},n_{2},\dots,n_{m}}$. For any integer $\lambda>0$, $\lambda G$ denotes the graph consisting of $\lambda$ edge-disjoint copies of $G$.

For any two graphs $G$ and $H$ of orders $m$ and $n$, respectively, the corona product $G\odot H$ is the graph obtained by taking one copy of $G$ and $m$ copies of $H$ such that the $i$th vertex of $G$ is connected to every vertex in the $i$th copy of $H$. We define the sunlet graph $L_{2k}$ with $V(L_{2k})$ $=$ $\left\{x_{1},x_{2},\dots,x_{k},x_{k+1},x_{k+2},\dots,x_{2k}\right\}$ and $E(L_{2k})$ $=$ $\left\{x_{i}x_{i+1}\cup x_{i}x_{k+i}\ |\ i=1,2,...,k\ \textnormal{and subscripts of the first term is taken addition modulo}\ k\right\}$. We denote it by

$L_{2k}$ $=$ $\begin{pmatrix}x_{1}&x_{2}&\dots&x_{k}\\ x_{k+1}&x_{k+2}&\dots&x_{2k}\end{pmatrix}$.

Clearly, $L_{2k}\cong C_{k}\odot K_{1}$.

The Cartesian product of two graphs, $G$ and $H$, denotes by $G\Box H$, has the vertex set $V(G)\times V(H)$ and in which two vertices $(g,h)$ and $(g^{\prime},h^{\prime})$ are adjacent if and only if either $g=g^{\prime}$ and $h$ is adjacent to $h^{\prime}$ in $H$ or $h=h^{\prime}$ and $g$ is adjacent to $g^{\prime}$ in $G$. It is well known that Cartesian product is commutative, associative and distributive over edge-disjoint union of graphs.

We shall use the following notation throughout the paper. Let $G$ and $H$ be simple graphs with vertex sets $V(G)$ $=$ $\left\{x_{1},x_{2},\dots,x_{n}\right\}$ and $V(H)$ $=$ $\left\{y_{1},y_{2},\dots,y_{m}\right\}$. Then for our convenience, we write $V(G)\times V(H)$ $=$ $\bigcup_{i=1}^{n}$ $X_{i}$, where $X_{i}$ stands for $x_{i}\times V(H)$. Further, in the sequel, we shall denote the vertices of $X_{i}$ as $\left\{x_{i}^{j}|1\leq j\leq m\right\}$, where $x_{i}^{j}$ stands for the vertex $(x_{i},y_{j})\in V(G)\times V(H)$.

By a decomposition of a graph $G$, we mean a list of edge-disjoint subgraphs whose union is $G$. For a graph $G$, if $E(G)$ can be partitioned into $E_{1},E_{2},...,E_{k}$ such that the subgraph induced by $E_{i}$ is $H_{i}$, for all $i$, $1\leq i\leq k$, then we say that $H_{1},H_{2},...,H_{k}$ decompose $G$ and we write $G$ $=$ $H_{1}\ \oplus\ H_{2}\ \oplus...\ \oplus\ H_{k}$, since $H_{1},H_{2},...,H_{k}$ are edge-disjoint subgraphs of $G$. For $1\leq i\leq k$, if $H_{i}$ $=$ $H$, we say that $G$ has a $H$- decomposition.

Study of $H$-decomposition of graphs is not new. Many authors around the world are working in the field of cycle decomposition, path decomposition, star decompositon, trail decomposition, Hamilton cycle decomposition problems in graphs. Here we consider the sunlet decomposition of product graphs. Anitha and Lekshmi [2, 3] proved that $n$-sun decomposition of complete graph, complete bipartite graph and the Harary graphs. Liang and Guo [8, 9] gave the existence spectrum of a $k$-sun system of order $v$ as $k=2,4,5,6,8$. Fu et. al. [5, 6] obtained that $3$-sun decompositions of $K_{p,p,r}$, $K_{n}$and embed a cyclic steiner triple system of order $n$ into a $3$-sun system of order $2n-1$, for $n=1\pmod{6}$. Further they obtained $k$-sun system when $k=6,10,14,2^{t},$ for $t>1$. Fu et. al. [4] obtained the existence of a $5$-sun system of order $v$. Gionfriddo et.al. [7] obtained the spectrum for uniformly resolvable decompositions of $K_{v}$ into $1$-factor and $h$-suns. Akwu and Ajayi [1] obtained the necessary and sufficient conditions for the existence of decomposition of $K_{n}\otimes\overline{K_{m}}$ and $(K_{n}-I)\otimes\overline{K_{m}}$, where $I$ denote the $1$-factor of a complete graph into sunlet graph of order twice the prime. Sowndhariya and Muthusamy [10] obtained necessary and sufficient conditions for the existence of decomposition of $K_{m}\times K_{n}$ and $K_{m}\otimes\overline{K_{n}}$ into sunlet graph of order eight.

In this paper, we prove the existence of $L_{8}$-decomposition of $K_{m}\Box K_{n}$. In fact, we establish necessary and sufficient conditions for the existence of $L_{8}$-decomposition of $K_{m}\Box K_{n}$. To prove our results, we state the following:

Theorem 1.1** ([10]).**

For $k\geq 3$, $K_{4k+1}$ has a $L_{2k}$- decomposition.

Theorem 1.2** ([10]).**

For an integer $p>0$, $K_{16p+1}$ has a $L_{8}$- decomposition.

Lemma 1.1** ([10]).**

There exists a $L_{8}$- decomposition of $K_{16}$.

Theorem 1.3** ([10]).**

For any $m,n\geq 4$, $K_{m,n}$ has a $L_{8}$- decomposition if and only if $mn\equiv 0\ (mod\ 8)$ except $(m,n)=(8,5),(4,4r+2)$ where $r>0$.

2 Decomposition of $K_{m}\Box K_{n}$ into Sunlet Graph of order Eight

2.1 Necessary Conditions

Lemma 2.1**.**

If $K_{m}\Box K_{n}$has an $L_{8}$-decomposition, then either

$m,n\equiv\ 0\ (mod\ 4)$** 2. 2.

$m\equiv\ 0\ (mod\ 8),\ n\equiv\ 0\ (mod\ 2)$** 3. 3.

$m\equiv\ 0\ (mod\ 16)$** 4. 4.

$m\equiv\ 1\ (mod\ 16),\ n\equiv\ 1\ (mod\ 16)$** 5. 5.

$m\equiv\ 15\ (mod\ 16),\ n\equiv\ 3\ (mod\ 16)$** 6. 6.

$m\equiv\ 13\ (mod\ 16),\ n\equiv\ 5\ (mod\ 16)$** 7. 7.

$m\equiv\ 11\ (mod\ 16),\ n\equiv\ 7\ (mod\ 16)$** 8. 8.

$m\equiv\ 9\ (mod\ 16),\ n\equiv\ 9\ (mod\ 16)$**

Proof.

The graph $K_{m}\Box K_{n}$ has $mn$ vertices, each having degree $m+n-2$. Hence, $K_{m}\Box K_{n}$ has $\frac{mn(m+n-2)}{2}$ edges. Assume that $K_{m}\Box K_{n}$ admits an $L_{8}$- decomposition. Then the number of edges in the graph must be divisible by $8$. i.e., $16|mn(m+n-2)$. Hence these conditions are met in each of the above eight cases and only in these cases. ∎

2.2 Sufficient Conditions

We now prove the above necessary conditions are also sufficiency.

Lemma 2.2**.**

If $m\equiv 0\pmod{4}$ and $n\equiv 0\pmod{4}$, then the graph $K_{m}\Box K_{n}$ has an $L_{8}$-decomposition.

Proof.

Let $m=4s$ and $n=4t$ for some $s,t>0$. We can divide the graph $K_{m}\Box K_{n}$ into $st\left(K_{4}\Box K_{4}\right)$ and the remaining graph can be viewed as $2st(s+t-2)K_{4,4}$. The $L_{8}$-decomposition of $K_{4}\Box K_{4}$ is shown in Fig 1. By Theorem 1.3, $K_{4,4}$ has an $L_{8}$-decomposititon. Hence the graph $K_{m}\Box K_{n}$ has the desired decomposition. ∎

Lemma 2.3**.**

If $m\equiv 0\pmod{8}$ and $n\equiv 0\pmod{2}$, then the graph $K_{m}\Box K_{n}$ has an $L_{8}$-decomposition.

Proof.

Let $m\equiv 0\pmod{8}$ and $n\equiv 0\ (or)\ 4\pmod{8}$, then the proof follows from Lemma 2.2. Now let $m=0\pmod{8}$ and consider two cases for $n$.

Case (1) $n\equiv\ 2\pmod{8}$.

Let $m=8s$ and $n=8t+2$ for some $s,t>0$. Then we can write $K_{m}\Box K_{n}$ as $s(t-1)\left(K_{8}\Box K_{8}\right)$ $\oplus$ $s\left(K_{8}\Box K_{10}\right)$ $\oplus$ $\frac{s\left[(t-1)(t-2)+(8t+2)(s-1)\right]}{2}K_{8,8}$ $\oplus$ $8s(t-1)K_{8,10}$. An $L_{8}$-decomposition of $K_{8}\Box K_{10}$ is presented in Appendix $4.1.1$ and the $L_{8}$-decomposition of the graphs $K_{8}\Box K_{8}$, $K_{8,8}$ and $K_{8,10}$ follows from Lemma 2.2 and Theorem 1.3.

Case (2) $n\equiv\ 6\pmod{8}$.

Let $m=8s$ and $n=8t+6$ for some $s,t>0$. Then we can write $K_{m}\Box K_{n}$ as $st\left(K_{8}\Box K_{8}\right)$ $\oplus$ $s\left(K_{8}\Box K_{6}\right)$ $\oplus$ $s\left[(s-1)(4t+3)+4t(t-1)\right]K_{8,8}$ $\oplus$ $8stK_{8,6}$. An $L_{8}$-decomposition of $K_{8}\Box K_{6}$ is appears in Appendix $4.1.2$ and the $L_{8}$-decomposition of the remaining graphs follows from Lemma 2.2 and Theorem 1.3. Hence the graph $K_{m}\Box K_{n}$ has the desired decomposition. ∎

Lemma 2.4**.**

If $m\equiv 0\pmod{16}$, then the graph $K_{m}\Box K_{n}$ has an $L_{8}$-decomposition.

Proof.

Let $m\equiv 0\pmod{16}$ and $n\equiv 0,2,4,6\pmod{8}$, then the proof follows from Lemma 2.3. Let $m=0\pmod{16}$ and consider four cases for odd $n$.

Case (1) $n\equiv\ 1\pmod{8}$.

Let $m=16s$ and $n=8t+1$ for some $s,t>0$. Then we can write $K_{m}\Box K_{n}$ as $s(t-1)\left(K_{16}\Box K_{8}\right)$ $\oplus$ $s\left(K_{16}\Box K_{9}\right)$ $\oplus$ $8s(t-1)(t-2)K_{8,8}$ $\oplus$ $\frac{s(8t+1)(s-1)}{2}K_{16,16}$ $\oplus$ $16s(t-1)K_{8,9}$. An $L_{8}$-decomposition of $K_{16}\Box K_{9}$ is shown in Appendix $4.2.1$ and the $L_{8}$-decomposition of the remaining graphs follows from Lemma 2.2 and Theorem 1.3.

Case (2) $n\equiv\ 3\pmod{8}$.

Let $m=16s$ and $n=8t+3$ for some $s,t>0$. Then we can write $K_{m}\Box K_{n}$ as $s(t-1)\left(K_{16}\Box K_{8}\right)$ $\oplus$ $s\left(K_{16}\Box K_{11}\right)$ $\oplus$ $8s(t-1)(t-2)K_{8,8}$ $\oplus$ $\frac{s(8t+3)(s-1)}{2}K_{16,16}$ $\oplus$ $16s(t-1)K_{8,11}$. An $L_{8}$-decomposition of $K_{16}\Box K_{11}$ is shown in Appendix $4.2.2$ and the $L_{8}$-decomposition of the remaining graphs follows from Lemma 2.2 and Theorem 1.3.

Case (3) $n\equiv\ 5\pmod{8}$.

Let $m=16s$ and $n=8t+5$ for some $s,t>0$. Then we can write $K_{m}\Box K_{n}$ as $s(t-1)\left(K_{16}\Box K_{8}\right)$ $\oplus$ $s\left(K_{16}\Box K_{13}\right)$ $\oplus$ $8s(t-1)(t-2)K_{8,8}$ $\oplus$ $\frac{s(8t+5)(s-1)}{2}K_{16,16}$ $\oplus$ $16s(t-1)K_{8,13}$. An $L_{8}$-decomposition of $K_{16}\Box K_{13}$ is presented in Appendix $4.2.3$ and the $L_{8}$-decomposition of the remaining graphs follows from Lemma 2.2 and Theorem 1.3.

Case (4) $n\equiv\ 7\pmod{8}$.

Let $m=16s$ and $n=8t+7$ for some $s,t>0$. Then we can write $K_{m}\Box K_{n}$ as $st\left(K_{16}\Box K_{8}\right)$ $\oplus$ $s\left(K_{16}\Box K_{7}\right)$ $\oplus$ $8st(t-1)K_{8,8}$ $\oplus$ $\frac{s(8t+7)(s-1)}{2}K_{16,16}$ $\oplus$ $16stK_{8,7}$. An $L_{8}$-decomposition of $K_{16}\Box K_{7}$ is presented in Appendix $4.2.4$ and the $L_{8}$-decomposition of the remaining graphs follows from Lemma 2.2 and Theorem 1.3.

Hence the graph $K_{m}\Box K_{n}$ has the desired decomposition. ∎

Lemma 2.5**.**

If $m\equiv 1\pmod{16}$ and $n\equiv 1\pmod{1}6$, then the graph $K_{m}\Box K_{n}$ has an $L_{8}$-decomposition.

Proof.

Let $m=16s+1$ and $n=16t+1$ for some $s,t>0$. Then we can write $K_{m}\Box K_{n}$ as $(16t+1)K_{16s+1}$ $\oplus$ $(16s+1)K_{16t+1}$. By Theorem 1.2, the graph $K_{m}\Box K_{n}$ has the desired decomposition. ∎

Lemma 2.6**.**

If $m\equiv 15\pmod{16}$ and $n\equiv 3\pmod{16}$, then the graph $K_{m}\Box K_{n}$ has an $L_{8}$-decomposition.

Proof.

Let $m=16s+15$ and $n=16t+3$ for some $s,t>0$. We can write $K_{m}\Box K_{n}$ as $(16t+3)K_{16s+15}$ $\oplus$ $(16s+15)K_{16t+1}$. Now the first $16t$ columns can be viewed as $K_{16s}\ \oplus\ K_{15}\oplus sK_{16,15}$ and the first $16s$ rows can be viewed as $K_{16(t-1)}\ \oplus K_{19}\ \oplus(t-1)K_{16,19}$. Then $K_{16s}(=sK_{16}\ \oplus\ \frac{s(s-1)}{2}K_{16,16})$, $K_{16(t-1)}(=(t-1)K_{16}\oplus\frac{(t-1)(t-2)}{2}K_{16,16})$, $K_{16,15}$ and $K_{16,19}$ has an $L_{8}$-decomposition by Lemma 1.1 and Theorem 1.3. The graph $K_{19}$ can be viewed as $K_{19}\backslash K_{3}\ \oplus\ K_{3}$. The $L_{8}$-decomposition of $K_{19}\backslash K_{3}$ follows from Appendix $4.3.1$. Then $16s\left(K_{19}\backslash K_{3}\right)$ has an $L_{8}$-decomposition. Then the remaining graph can be viewed as $s(K_{16}\Box K_{3})$, $t(K_{15}\Box K_{16})$ and $K_{15}\Box K_{3}$. Hence the desired decomposition follows from Appendixes ${4.3.2}$, ${4.3.3}$ and Lemma 2.4 ∎

Lemma 2.7**.**

If $m\equiv 13\pmod{16}$ and $n\equiv 5\pmod{16}$, then the graph $K_{m}\Box K_{n}$ has an $L_{8}$-decomposition.

Proof.

Let $m=16s+13$ and $n=16t+5$ for some $s,t>0$. Then we can write $K_{m}\Box K_{n}$ as $st\left(K_{16}\Box K_{16}\right)$ $\oplus$ $t\left(K_{16}\Box K_{13}\right)$ $\oplus$ $s\left(K_{16}\Box K_{5}\right)$ $\oplus$ $K_{13}\Box K_{5}$ $\oplus$ $\frac{t(t-1)(16s+13)+s(s-1)(16t+5)}{2}K_{16,16}$ $\oplus$ $s(16t+5)K_{16,13}$ $\oplus$ $t(16s+13)K_{16,5}$. An $L_{8}$-decomposition of $K_{16}\Box K_{5}$ and $K_{13}\Box K_{5}$ are presented in Appendixes $4.4.1$ and $4.4.2$, respectively and the $L_{8}$-decomposition of the remaining graphs follows from Lemma 2.4 and Theorem 1.3. Hence the graph $K_{m}\Box K_{n}$ has the desired decomposition. ∎

Lemma 2.8**.**

If $m\equiv 11\pmod{16}$ and $n\equiv 7\pmod{16}$, then the graph $K_{m}\Box K_{n}$ has an $L_{8}$-decomposition.

Proof.

Let $m=16s+11$ and $n=16t+7$ for some $s,t>0$. Then we can write $K_{m}\Box K_{n}$ as $st\left(K_{16}\Box K_{16}\right)$ $\oplus$ $t\left(K_{16}\Box K_{11}\right)$ $\oplus$ $s\left(K_{16}\Box K_{7}\right)$ $\oplus$ $K_{11}\Box K_{7}$ $\oplus$ $\frac{s(s-1)(16t+7)+t(t-1)(16s+11)}{2}K_{16,16}$ $\oplus$ $(16t+7)sK_{16,11}$ $\oplus$ $(16s+11)tK_{16,7}$. An $L_{8}$-decomposition of $K_{11}\Box K_{7}$ is presented in Appendix $4.5.1$ and the $L_{8}$-decomposition of the remaining graphs follows from Lemma 2.4 and Theorem 1.3. Hence the graph $K_{m}\Box K_{n}$ has the desired decomposition. ∎

Lemma 2.9**.**

If $m\equiv 9\pmod{16}$ and $n\equiv 9\pmod{16}$, then the graph $K_{m}\Box K_{n}$ has an $L_{8}$-decomposition.

Proof.

Let $m=16s+9$ and $n=16t+9$ for some $s,t>0$. Then we can write $K_{m}\Box K_{n}$ as $st\left(K_{16}\Box K_{16}\right)$ $\oplus$ $(s+t)\left(K_{16}\Box K_{9}\right)$ $\oplus$ $K_{9}\Box K_{9}$ $\oplus$ $\frac{s(s-1)(16t+9)+t(t-1)(16s+9)}{2}K_{16,16}$ $\oplus$ $\left[s(16t+9)+t(16s+9)\right]K_{16,9}$. An $L_{8}$-decomposition of $K_{11}\Box K_{7}$ is presented in Appendix $4.6.1$ and the $L_{8}$-decomposition of the remaining graphs follows from Lemma 2.4 and Theorem 1.3. Hence the graph $K_{m}\Box K_{n}$ has the desired decomposition. ∎

2.3 Main Theorem

Combining the results from Lemma 2.1 to Lemma 2.9, we get the following main results.

Theorem 2.1**.**

The graph $K_{m}\Box K_{n}$ admits an $L_{8}$- decomposition if and only if one of the following holds:

$m,n\equiv\ 0\ (mod\ 4)$** 2. 2.

$m\equiv\ 0\ (mod\ 8),\ n\equiv\ 0\ (mod\ 2)$** 3. 3.

$m\equiv\ 0\ (mod\ 16)$** 4. 4.

$m\equiv\ 1\ (mod\ 16),\ n\equiv\ 1\ (mod\ 16)$** 5. 5.

$m\equiv\ 15\ (mod\ 16),\ n\equiv\ 3\ (mod\ 16)$** 6. 6.

$m\equiv\ 13\ (mod\ 16),\ n\equiv\ 5\ (mod\ 16)$** 7. 7.

$m\equiv\ 11\ (mod\ 16),\ n\equiv\ 7\ (mod\ 16)$** 8. 8.

$m\equiv\ 9\ (mod\ 16),\ n\equiv\ 9\ (mod\ 16)$**

3 Acknowledgments:

The first author thank the Department of Science and Technology, Government of India, New Delhi for its financial support through the Grant No.DST/INSPIRE Fellowship/2015/IF150211. The second author thank the University Grant Commission, Government of India, New Delhi for its support through the grant No.F.510/7/DRS-I/2016(SAP-I).

4 Appendix

4.1 $L_{8}$-decomposition required for Lemma 2.3

4.1.1 An $L_{8}$- decomposition of $K_{8}\Box K_{10}$

$\begin{pmatrix}x_{i}^{1}&x_{i}^{6}&x_{i}^{2}&x_{i}^{7}\\ x_{i}^{3}&x_{i}^{8}&x_{i}^{4}&x_{i}^{9}\end{pmatrix}$, $\begin{pmatrix}x_{i}^{3}&x_{i}^{8}&x_{i}^{4}&x_{i}^{9}\\ x_{i}^{10}&x_{i}^{5}&x_{i}^{7}&x_{i}^{2}\end{pmatrix}$, $\begin{pmatrix}x_{i}^{2}&x_{i}^{3}&x_{i}^{4}&x_{i}^{5}\\ x_{i}^{1}&x_{i}^{7}&x_{i}^{6}&x_{i}^{9}\end{pmatrix}$ for $i=1,2,...,8$;

$\begin{pmatrix}x_{1}^{j}&x_{5}^{j}&x_{2}^{j}&x_{6}^{j}\\ x_{3}^{j}&x_{7}^{j}&x_{4}^{j}&x_{8}^{j}\end{pmatrix}$, $\begin{pmatrix}x_{3}^{j}&x_{7}^{j}&x_{4}^{j}&x_{8}^{j}\\ x_{5}^{j}&x_{1}^{j}&x_{6}^{j}&x_{2}^{j}\end{pmatrix}$ for $j=1,2,...,10$;

$\begin{pmatrix}x_{1}^{j}&x_{2}^{j}&x_{3}^{j}&x_{4}^{j}\\ x_{8}^{j}&x_{7}^{j}&x_{6}^{j}&x_{5}^{j}\end{pmatrix}$ for $j=1,3,5,7,8,9$; $\begin{pmatrix}x_{5}^{j}&x_{6}^{j}&x_{7}^{j}&x_{8}^{j}\\ x_{4}^{j}&x_{3}^{j}&x_{2}^{j}&x_{1}^{j}\end{pmatrix}$ for $j=2,4,6,10$;

$\begin{pmatrix}x_{i}^{1}&x_{i}^{5}&x_{i}^{6}&x_{i}^{10}\\ x_{i}^{8}&x_{i}^{k_{1}}&x_{i}^{k_{2}}&x_{i}^{k_{3}}\end{pmatrix}$ for $i=1,2,3,4,\ (k_{1},k_{2},k_{3})=(3,9,4)$ & $i=5,6,7,8,\\ (k_{1},k_{2},k_{3})=(7,3,2)$;

$\begin{pmatrix}x_{i}^{7}&x_{i}^{8}&x_{i}^{9}&x_{i}^{10}\\ x_{i}^{6}&x_{i}^{2}&x_{i}^{1}&x_{i}^{k}\end{pmatrix}$ for $i=1,2,3,4,\ k=5$ & $i=5,6,7,8,\ k=4$;

$\begin{pmatrix}x_{1}^{j}&x_{2}^{j}&x_{3}^{j}&x_{4}^{j}\\ x_{1}^{k}&x_{2}^{k}&x_{3}^{k}&x_{4}^{k}\end{pmatrix}$ for $(j,k)=(2,10),(4,1),(6,3),(10,8)$;

$\begin{pmatrix}x_{5}^{j}&x_{6}^{j}&x_{7}^{j}&x_{8}^{j}\\ x_{5}^{k}&x_{6}^{k}&x_{7}^{k}&x_{8}^{k}\end{pmatrix}$ for $(j,k)=(1,4),(3,5),(5,10),(7,5),(8,10),(9,6)$.

4.1.2 An $L_{8}$- decomposition of $K_{8}\Box K_{6}$

$\begin{pmatrix}x_{1}^{j}&x_{5}^{j}&x_{2}^{j}&x_{6}^{j}\\ x_{1}^{k}&x_{7}^{j}&x_{2}^{k}&x_{8}^{j}\end{pmatrix}$, $\begin{pmatrix}x_{3}^{j}&x_{7}^{j}&x_{4}^{j}&x_{8}^{j}\\ x_{3}^{k}&x_{1}^{j}&x_{4}^{k}&x_{2}^{j}\end{pmatrix}$ for $(j,k)=(1,3),(5,2),(6,4)$;

$\begin{pmatrix}x_{1}^{j}&x_{5}^{j}&x_{2}^{j}&x_{6}^{j}\\ x_{3}^{j}&x_{5}^{k}&x_{4}^{j}&x_{6}^{k}\end{pmatrix}$; $\begin{pmatrix}x_{3}^{j}&x_{7}^{j}&x_{4}^{j}&x_{8}^{j}\\ x_{5}^{j}&x_{7}^{k}&x_{6}^{j}&x_{8}^{k}\end{pmatrix}$ for $(j,k)=(2,5),(3,1),(4,6)$;

$\begin{pmatrix}x_{i}^{1}&x_{i}^{2}&x_{i}^{3}&x_{i}^{4}\\ x_{k}^{1}&x_{9-i}^{2}&x_{9-i}^{3}&x_{9-i}^{4}\end{pmatrix}$ for $(i,k)=(1,3),(2,4),(3,6)$;

$\begin{pmatrix}x_{i}^{1}&x_{i}^{5}&x_{i}^{3}&x_{i}^{6}\\ x_{i+1}^{1}&x_{k}^{5}&x_{i+1}^{3}&x_{k}^{6}\end{pmatrix}$ for $(i,k)=(1,8),(2,4),(3,5)$;

$\begin{pmatrix}x_{i}^{1}&x_{i}^{2}&x_{i}^{3}&x_{i}^{4}\\ x_{i+1}^{1}&x_{i+1}^{2}&x_{i+1}^{3}&x_{i+1}^{4}\end{pmatrix}$ for $i=4,5,6,7$; $\begin{pmatrix}x_{i}^{2}&x_{i}^{4}&x_{i}^{5}&x_{i}^{6}\\ x_{i+1}^{2}&x_{i+1}^{4}&x_{i+1}^{5}&x_{i+1}^{6}\end{pmatrix}$ for $i=1,2,3$;

$\begin{pmatrix}x_{8}^{1}&x_{8}^{2}&x_{8}^{3}&x_{8}^{4}\\ x_{5}^{1}&x_{5}^{2}&x_{5}^{3}&x_{5}^{4}\end{pmatrix}$, $\begin{pmatrix}x_{4}^{1}&x_{4}^{5}&x_{4}^{3}&x_{4}^{6}\\ x_{1}^{1}&x_{6}^{5}&x_{1}^{3}&x_{6}^{6}\end{pmatrix}$, $\begin{pmatrix}x_{5}^{1}&x_{5}^{5}&x_{5}^{3}&x_{5}^{6}\\ x_{3}^{1}&x_{6}^{5}&x_{7}^{3}&x_{6}^{6}\end{pmatrix}$, $\begin{pmatrix}x_{6}^{1}&x_{6}^{5}&x_{6}^{3}&x_{6}^{6}\\ x_{4}^{1}&x_{7}^{5}&x_{8}^{3}&x_{7}^{6}\end{pmatrix}$,

$\begin{pmatrix}x_{7}^{1}&x_{7}^{5}&x_{7}^{3}&x_{7}^{6}\\ x_{2}^{1}&x_{8}^{5}&x_{1}^{3}&x_{8}^{6}\end{pmatrix}$, $\begin{pmatrix}x_{8}^{1}&x_{8}^{5}&x_{8}^{3}&x_{8}^{6}\\ x_{1}^{1}&x_{5}^{5}&x_{2}^{3}&x_{5}^{6}\end{pmatrix}$, $\begin{pmatrix}x_{4}^{2}&x_{4}^{4}&x_{4}^{5}&x_{4}^{6}\\ x_{1}^{2}&x_{1}^{4}&x_{1}^{5}&x_{1}^{6}\end{pmatrix}$, $\begin{pmatrix}x_{5}^{2}&x_{5}^{4}&x_{5}^{5}&x_{5}^{6}\\ x_{7}^{2}&x_{7}^{4}&x_{4}^{5}&x_{4}^{6}\end{pmatrix}$,

$\begin{pmatrix}x_{6}^{2}&x_{6}^{4}&x_{6}^{5}&x_{6}^{6}\\ x_{8}^{2}&x_{8}^{4}&x_{3}^{5}&x_{3}^{6}\end{pmatrix}$, $\begin{pmatrix}x_{7}^{2}&x_{7}^{4}&x_{7}^{5}&x_{7}^{6}\\ x_{1}^{2}&x_{1}^{4}&x_{2}^{5}&x_{2}^{6}\end{pmatrix}$, $\begin{pmatrix}x_{8}^{2}&x_{8}^{4}&x_{8}^{5}&x_{8}^{6}\\ x_{2}^{2}&x_{2}^{4}&x_{1}^{5}&x_{1}^{6}\end{pmatrix}$.

4.2 $L_{8}$-decomposition required for Lemma 2.4

4.2.1 An $L_{8}$- decomposition of $K_{16}\Box K_{9}$

$\begin{pmatrix}x_{1}^{j}&x_{2}^{j}&x_{15}^{j}&x_{16}^{j}\\ x_{12}^{j}&x_{4}^{j}&x_{6}^{j}&x_{5}^{j}\end{pmatrix}$, $\begin{pmatrix}x_{3}^{j}&x_{4}^{j}&x_{13}^{j}&x_{14}^{j}\\ x_{12}^{j}&x_{11}^{j}&x_{5}^{j}&x_{15}^{j}\end{pmatrix}$ for $j=1,4,5,7,8$;

$\begin{pmatrix}x_{5}^{j}&x_{6}^{j}&x_{11}^{j}&x_{12}^{j}\\ x_{15}^{j}&x_{7}^{j}&x_{2}^{j}&x_{16}^{j}\end{pmatrix}$, $\begin{pmatrix}x_{7}^{j}&x_{8}^{j}&x_{9}^{j}&x_{10}^{j}\\ x_{13}^{j}&x_{15}^{j}&x_{11}^{j}&x_{1}^{j}\end{pmatrix}$ for $j=1,4,5,7,8$;

$\begin{pmatrix}x_{1}^{j}&x_{13}^{j}&x_{2}^{j}&x_{14}^{j}\\ x_{11}^{j}&x_{6}^{j}&x_{10}^{j}&x_{5}^{j}\end{pmatrix}$, $\begin{pmatrix}x_{3}^{j}&x_{15}^{j}&x_{4}^{j}&x_{16}^{j}\\ x_{11}^{j}&x_{1}^{j}&x_{10}^{j}&x_{2}^{j}\end{pmatrix}$ for $j=1,3,4,5,6,7,8$;

$\begin{pmatrix}x_{1}^{j}&x_{7}^{j}&x_{2}^{j}&x_{8}^{j}\\ x_{4}^{j}&x_{16}^{j}&x_{9}^{j}&x_{13}^{j}\end{pmatrix}$, $\begin{pmatrix}x_{9}^{j}&x_{15}^{j}&x_{10}^{j}&x_{16}^{j}\\ x_{1}^{j}&x_{12}^{j}&x_{3}^{j}&x_{14}^{j}\end{pmatrix}$ for $j=1,3,4,5,6,7,8,9$;

$\begin{pmatrix}x_{1}^{j}&x_{5}^{j}&x_{2}^{j}&x_{6}^{j}\\ x_{3}^{j}&x_{8}^{j}&x_{12}^{j}&x_{16}^{j}\end{pmatrix}$, $\begin{pmatrix}x_{9}^{j}&x_{13}^{j}&x_{10}^{j}&x_{14}^{j}\\ x_{3}^{j}&x_{16}^{j}&x_{12}^{j}&x_{7}^{j}\end{pmatrix}$ for $j=1,3,4,5,7,8$;

$\begin{pmatrix}x_{7}^{j}&x_{11}^{j}&x_{8}^{j}&x_{12}^{j}\\ x_{9}^{j}&x_{15}^{j}&x_{10}^{j}&x_{6}^{j}\end{pmatrix}$ for $j=1,3,4,5,6,7,8$; $\begin{pmatrix}x_{11}^{j}&x_{13}^{j}&x_{12}^{j}&x_{14}^{j}\\ x_{16}^{j}&x_{15}^{j}&x_{4}^{j}&x_{8}^{j}\end{pmatrix}$ for $j=1,3,4,5,7,8$;

$\begin{pmatrix}x_{5}^{j}&x_{9}^{j}&x_{6}^{j}&x_{10}^{j}\\ x_{7}^{j}&x_{12}^{j}&x_{8}^{j}&x_{11}^{j}\end{pmatrix}$ for $j=1,2,3,4,5,6,7,8$; $\begin{pmatrix}x_{1}^{j}&x_{5}^{j}&x_{2}^{j}&x_{6}^{j}\\ x_{3}^{j}&x_{16}^{j}&x_{12}^{j}&x_{14}^{j}\end{pmatrix}$ for $j=6,9$;

$\begin{pmatrix}x_{3}^{j}&x_{7}^{j}&x_{4}^{j}&x_{8}^{j}\\ x_{13}^{j}&x_{15}^{j}&x_{14}^{j}&x_{16}^{j}\end{pmatrix}$ for $j=1,2,...,8,9$; $\begin{pmatrix}x_{i}^{3}&x_{i}^{7}&x_{i}^{4}&x_{i}^{9}\\ x_{i}^{1}&x_{i}^{5}&x_{i}^{6}&x_{i}^{8}\end{pmatrix}$ for $i=1,2,...,15,16$;

$\begin{pmatrix}x_{i}^{9}&x_{i+4}^{9}&x_{i+1}^{9}&x_{i+5}^{9}\\ x_{i+2}^{9}&x_{i+4}^{5}&x_{i+3}^{9}&x_{i+5}^{5}\end{pmatrix}$ for $i=5,7$; $\begin{pmatrix}x_{i}^{1}&x_{i}^{8}&x_{i}^{2}&x_{i}^{9}\\ x_{i}^{5}&x_{i}^{6}&x_{i}^{4}&x_{i}^{7}\end{pmatrix}$ for $i=1,2,...,7,8$;

$\begin{pmatrix}x_{i}^{j}&x_{12+i}^{j}&x_{i+1}^{j}&x_{13+i}^{j}\\ x_{11}^{j}&x_{12+i}^{5}&x_{10}^{j}&x_{13+i}^{5}\end{pmatrix}$ for $j=2,9\ \&\ i=1,3$;

$\begin{pmatrix}x_{3}^{j}&x_{5}^{j}&x_{4}^{j}&x_{6}^{j}\\ x_{2}^{j}&x_{11}^{j}&x_{9}^{j}&x_{k}^{j}\end{pmatrix}$ for $j=1,3,4,5,7,8,\ k=14\ \&\ j=2,6,9,\ k=16$;

$\begin{pmatrix}x_{i}^{5}&x_{i}^{6}&x_{i}^{7}&x_{i}^{8}\\ x_{i}^{4}&x_{i}^{1}&x_{i}^{k}&x_{i}^{3}\end{pmatrix}$ for $i=1,2,...,7,8,\ k=2,\ \&\ i=9,...,16,\ k=9$;

$\begin{pmatrix}x_{5}^{j}&x_{6}^{j}&x_{11}^{j}&x_{12}^{j}\\ x_{15}^{j}&x_{7}^{j}&x_{11}^{k}&x_{16}^{j}\end{pmatrix}$, $\begin{pmatrix}x_{11}^{j}&x_{13}^{j}&x_{12}^{j}&x_{14}^{j}\\ x_{16}^{j}&x_{15}^{j}&x_{12}^{k}&x_{8}^{j}\end{pmatrix}$ for $(j,k)=(6,3),(9,6)$;

$\begin{pmatrix}x_{7}^{j}&x_{8}^{j}&x_{9}^{j}&x_{10}^{j}\\ x_{13}^{j}&x_{15}^{j}&x_{11}^{j}&x_{10}^{k}\end{pmatrix}$, $\begin{pmatrix}x_{3}^{j}&x_{4}^{j}&x_{13}^{j}&x_{14}^{j}\\ x_{12}^{j}&x_{11}^{j}&x_{13}^{k}&x_{15}^{j}\end{pmatrix}$ for $(j,k)=(2,7),(6,3),(9,6)$;

$\begin{pmatrix}x_{1}^{j}&x_{2}^{j}&x_{15}^{j}&x_{16}^{j}\\ x_{12}^{j}&x_{4}^{j}&x_{15}^{k}&x_{16}^{k}\end{pmatrix}$ for $(j,k)=(2,7),(3,5),(6,3),(9,6)$;

$\begin{pmatrix}x_{i}^{3}&x_{i+1}^{3}&x_{16-i}^{3}&x_{17-i}^{3}\\ x_{k_{1}}^{3}&x_{k_{2}}^{3}&x_{16-i}^{5}&x_{17-i}^{5}\end{pmatrix}$; for $(i,k_{1},k_{2})=(3,12,11)(5,15,7),(7,13,15)$;

$\begin{pmatrix}x_{9}^{j}&x_{13}^{j}&x_{10}^{j}&x_{14}^{j}\\ x_{9}^{k}&x_{16}^{j}&x_{12}^{j}&x_{14}^{k}\end{pmatrix}$ for $(j,k)=(2,7),(6,3),(9,6)$;

$\begin{pmatrix}x_{i}^{1}&x_{i}^{2}&x_{i}^{3}&x_{i}^{4}\\ x_{i}^{7}&x_{k}^{2}&x_{i}^{6}&x_{i}^{8}\end{pmatrix}$ for $(i,k)=(1,10),(2,11),(3,9),(4,12),(5,13),(6,15),(7,14),(8,5)$;

$\begin{pmatrix}x_{i}^{1}&x_{i}^{2}&x_{i}^{3}&x_{i}^{4}\\ x_{i}^{7}&x_{i}^{6}&x_{k}^{3}&x_{i}^{8}\end{pmatrix}$ for $(i,k)=(9,11),(10,1),(11,2),(12,16),(13,5),(14,15),\\ (15,6),(16,5)$;

$\begin{pmatrix}x_{i}^{1}&x_{i}^{8}&x_{i}^{2}&x_{i}^{9}\\ x_{i}^{5}&x_{i}^{6}&x_{i}^{4}&x_{k}^{9}\end{pmatrix}$ for $(i,k)=(9,12),(10,11),(11,15),(12,6),(13,6),(14,5),\\ (15,1),(16,2)$;

$\begin{pmatrix}x_{i}^{2}&x_{i}^{5}&x_{i}^{9}&x_{i}^{6}\\ x_{k_{1}}^{2}&x_{i}^{3}&x_{k_{2}}^{9}&x_{k_{2}}^{6}\end{pmatrix}$ for $(i,k_{1},k_{2})=(1,15,10),(2,16,11),(3,10,9),(4,1,12),(5,14,13),\\ (6,13,15),(7,6,14),(8,14,5)$;

$\begin{pmatrix}x_{7}^{2}&x_{11}^{2}&x_{8}^{2}&x_{12}^{2}\\ x_{9}^{2}&x_{15}^{2}&x_{10}^{2}&x_{12}^{5}\end{pmatrix}$, $\begin{pmatrix}x_{1}^{2}&x_{7}^{2}&x_{2}^{2}&x_{8}^{2}\\ x_{9}^{2}&x_{16}^{2}&x_{12}^{2}&x_{13}^{2}\end{pmatrix}$, $\begin{pmatrix}x_{9}^{2}&x_{15}^{2}&x_{10}^{2}&x_{16}^{2}\\ x_{9}^{5}&x_{12}^{2}&x_{10}^{5}&x_{11}^{2}\end{pmatrix}$, $\begin{pmatrix}x_{11}^{2}&x_{13}^{2}&x_{12}^{2}&x_{14}^{2}\\ x_{11}^{5}&x_{15}^{2}&x_{12}^{7}&x_{16}^{2}\end{pmatrix}$,

$\begin{pmatrix}x_{1}^{2}&x_{5}^{2}&x_{2}^{2}&x_{6}^{2}\\ x_{3}^{2}&x_{16}^{2}&x_{9}^{2}&x_{12}^{2}\end{pmatrix}$, $\begin{pmatrix}x_{5}^{2}&x_{6}^{2}&x_{11}^{2}&x_{12}^{2}\\ x_{15}^{2}&x_{14}^{2}&x_{11}^{7}&x_{16}^{2}\end{pmatrix}$.

4.2.2 An $L_{8}$- decomposition of $K_{16}\Box K_{11}$

$\begin{pmatrix}x_{5}^{j}&x_{6}^{j}&x_{11}^{j}&x_{12}^{j}\\ x_{15}^{j}&x_{7}^{j}&x_{2}^{j}&x_{16}^{j}\end{pmatrix}$, $\begin{pmatrix}x_{7}^{j}&x_{8}^{j}&x_{9}^{j}&x_{10}^{j}\\ x_{13}^{j}&x_{15}^{j}&x_{11}^{j}&x_{1}^{j}\end{pmatrix}$ for $j=1,2,4,5,7,8,10$;

$\begin{pmatrix}x_{1}^{j}&x_{13}^{j}&x_{2}^{j}&x_{14}^{j}\\ x_{11}^{j}&x_{6}^{j}&x_{10}^{j}&x_{5}^{j}\end{pmatrix}$, $\begin{pmatrix}x_{3}^{j}&x_{15}^{j}&x_{4}^{j}&x_{16}^{j}\\ x_{11}^{j}&x_{1}^{j}&x_{10}^{j}&x_{2}^{j}\end{pmatrix}$ for $j=2,5,6,7,8,9$;

$\begin{pmatrix}x_{1}^{j}&x_{7}^{j}&x_{2}^{j}&x_{8}^{j}\\ x_{4}^{j}&x_{16}^{j}&x_{9}^{j}&x_{13}^{j}\end{pmatrix}$, $\begin{pmatrix}x_{1}^{j}&x_{5}^{j}&x_{2}^{j}&x_{6}^{j}\\ x_{3}^{j}&x_{8}^{j}&x_{12}^{j}&x_{16}^{j}\end{pmatrix}$ for $j=1,2,4,5,...,10,11$;

$\begin{pmatrix}x_{3}^{j}&x_{7}^{j}&x_{4}^{j}&x_{8}^{j}\\ x_{13}^{j}&x_{15}^{j}&x_{14}^{j}&x_{16}^{j}\end{pmatrix}$, $\begin{pmatrix}x_{9}^{j}&x_{13}^{j}&x_{10}^{j}&x_{14}^{j}\\ x_{3}^{j}&x_{16}^{j}&x_{12}^{j}&x_{7}^{j}\end{pmatrix}$ for $j=1,2,3,5,...,11$;

$\begin{pmatrix}x_{i}^{3}&x_{i}^{7}&x_{i}^{4}&x_{i}^{10}\\ x_{i}^{11}&x_{i}^{6}&x_{i}^{8}&x_{i}^{9}\end{pmatrix}$, $\begin{pmatrix}x_{i}^{2}&x_{i}^{5}&x_{i}^{7}&x_{i}^{11}\\ x_{i}^{3}&x_{i}^{9}&x_{i}^{8}&x_{i}^{10}\end{pmatrix}$ for $i=1,2,5,6,7,8,10,11,12,14$;

$\begin{pmatrix}x_{i}^{2}&x_{i}^{5}&x_{i}^{7}&x_{i}^{11}\\ x_{i}^{1}&x_{i}^{9}&x_{i}^{8}&x_{i}^{10}\end{pmatrix}$, $\begin{pmatrix}x_{i}^{1}&x_{i}^{6}&x_{i}^{2}&x_{i}^{7}\\ x_{i}^{3}&x_{i}^{5}&x_{i}^{4}&x_{i}^{10}\end{pmatrix}$ for $i=15,16$;

$\begin{pmatrix}x_{i}^{3}&x_{i}^{4}&x_{i}^{9}&x_{i}^{8}\\ x_{i}^{5}&x_{i}^{6}&x_{i}^{11}&x_{i}^{10}\end{pmatrix}$, $\begin{pmatrix}x_{i}^{3}&x_{i}^{7}&x_{i}^{4}&x_{i}^{10}\\ x_{i}^{11}&x_{i}^{6}&x_{i}^{8}&x_{k}^{9}\end{pmatrix}$ for $i=15,16$;

$\begin{pmatrix}x_{1}^{j}&x_{2}^{j}&x_{15}^{j}&x_{16}^{j}\\ x_{1}^{k}&x_{2}^{k}&x_{6}^{j}&x_{5}^{j}\end{pmatrix}$, $\begin{pmatrix}x_{3}^{j}&x_{4}^{j}&x_{13}^{j}&x_{14}^{j}\\ x_{3}^{k}&x_{4}^{k}&x_{13}^{k}&x_{14}^{k}\end{pmatrix}$ for $(j,k)=(6,3),(9,3),(11,8)$;

$\begin{pmatrix}x_{5}^{j}&x_{6}^{j}&x_{11}^{j}&x_{12}^{j}\\ x_{5}^{k}&x_{6}^{k}&x_{11}^{k}&x_{12}^{k}\end{pmatrix}$, $\begin{pmatrix}x_{7}^{j}&x_{8}^{j}&x_{9}^{j}&x_{10}^{j}\\ x_{7}^{k}&x_{8}^{k}&x_{9}^{k}&x_{10}^{k}\end{pmatrix}$ for $(j,k)=(6,3),(9,3),(11,8)$;

$\begin{pmatrix}x_{1}^{j}&x_{2}^{j}&x_{15}^{j}&x_{16}^{j}\\ x_{12}^{j}&x_{4}^{j}&x_{6}^{j}&x_{5}^{j}\end{pmatrix}$ for $j=2,5,7,8,10$; $\begin{pmatrix}x_{3}^{j}&x_{5}^{j}&x_{4}^{j}&x_{6}^{j}\\ x_{2}^{j}&x_{11}^{j}&x_{9}^{j}&x_{14}^{j}\end{pmatrix}$ for $j=1,2,4,5,...,10,11$;

$\begin{pmatrix}x_{5}^{j}&x_{9}^{j}&x_{6}^{j}&x_{10}^{j}\\ x_{7}^{j}&x_{12}^{j}&x_{8}^{j}&x_{11}^{j}\end{pmatrix}$ for $j=2,3,...,10,11$; $\begin{pmatrix}x_{9}^{j}&x_{15}^{j}&x_{10}^{j}&x_{16}^{j}\\ x_{1}^{j}&x_{12}^{j}&x_{3}^{j}&x_{14}^{j}\end{pmatrix}$ for $j=6,7,8,9$;

$\begin{pmatrix}x_{3}^{j}&x_{4}^{j}&x_{13}^{j}&x_{14}^{j}\\ x_{12}^{j}&x_{11}^{j}&x_{5}^{j}&x_{15}^{j}\end{pmatrix}$ for $j=1,2,3,4,5,7,8,10$;

$\begin{pmatrix}x_{i}^{1}&x_{i}^{8}&x_{i}^{2}&x_{i}^{9}\\ x_{i}^{4}&x_{i}^{6}&x_{i}^{10}&x_{i}^{7}\end{pmatrix}$ for $i=1,2,5,6,12,14,15,16$;

$\begin{pmatrix}x_{7}^{j}&x_{11}^{j}&x_{8}^{j}&x_{12}^{j}\\ x_{9}^{j}&x_{15}^{j}&x_{10}^{j}&x_{6}^{j}\end{pmatrix}$ for $j=1,2,4,5,...,11$; $\begin{pmatrix}x_{i}^{5}&x_{i}^{10}&x_{i}^{6}&x_{i}^{11}\\ x_{k}^{5}&x_{i}^{1}&x_{i}^{9}&x_{i}^{8}\end{pmatrix}$ for $(i.k)=(15,12),(16,14)$;

$\begin{pmatrix}x_{9}^{j}&x_{15}^{j}&x_{10}^{j}&x_{16}^{j}\\ x_{9}^{k}&x_{12}^{j}&x_{3}^{j}&x_{14}^{j}\end{pmatrix}$ for $(j,k)=(1,10),(4,1),(10,11),(11,3)$;

$\begin{pmatrix}x_{3}^{j}&x_{15}^{j}&x_{4}^{j}&x_{16}^{j}\\ x_{3}^{k}&x_{1}^{j}&x_{4}^{k}&x_{2}^{j}\end{pmatrix}$ for $(j,k)=(10,11),(11,3)$; $\begin{pmatrix}x_{11}^{j}&x_{13}^{j}&x_{12}^{j}&x_{14}^{j}\\ x_{16}^{j}&x_{15}^{j}&x_{4}^{j}&x_{8}^{j}\end{pmatrix}$ for $j=1,2,3,5,...,11$;

$\begin{pmatrix}x_{1}^{j}&x_{2}^{j}&x_{15}^{j}&x_{16}^{j}\\ x_{12}^{j}&x_{4}^{j}&x_{15}^{k}&x_{16}^{k}\end{pmatrix}$ for $(j,k)=(1,11),(3,9),(4,5)$;

$\begin{pmatrix}x_{1}^{j}&x_{13}^{j}&x_{2}^{j}&x_{14}^{j}\\ x_{11}^{j}&x_{13}^{k}&x_{10}^{j}&x_{5}^{j}\end{pmatrix}$ for $(j,k)=(4,1),(10,11),(11,3)$;

$\begin{pmatrix}x_{3}^{j}&x_{15}^{j}&x_{4}^{j}&x_{16}^{j}\\ x_{3}^{k_{1}}&x_{15}^{k_{2}}&x_{4}^{k_{1}}&x_{16}^{k_{2}}\end{pmatrix}$ for $(j,k_{1},k_{2})=(1,10,5),(4,1,11)$;

$\begin{pmatrix}x_{i}^{1}&x_{i}^{6}&x_{i}^{2}&x_{i}^{7}\\ x_{i}^{3}&x_{k}^{6}&x_{i}^{4}&x_{i}^{10}\end{pmatrix}$ for $(i,k)=(1,12),(2,4),(3,12),(4,11),(5,15),(6,7),(7,13),(8,15),\\ (9,11),(10,1),(11,2),(13,5)$;

$\begin{pmatrix}x_{i}^{1}&x_{i}^{6}&x_{i}^{2}&x_{i}^{7}\\ x_{i}^{3}&x_{k_{1}}^{6}&x_{k_{2}}^{2}&x_{i}^{10}\end{pmatrix}$ for $(i,k_{1},k_{2})=(12,16,15),(14,15,16)$;

$\begin{pmatrix}x_{i}^{3}&x_{i}^{4}&x_{i}^{9}&x_{i}^{8}\\ x_{i}^{5}&x_{i}^{6}&x_{k}^{9}&x_{i}^{10}\end{pmatrix}$ for $(i,k)=(1,12),(2,4),(3,12),(4,11),(5,15),(6,7),(7,13),(8,15),\\ (9,11),(10,1),(11,2),(12,16),(13,5),(14,15)$;

$\begin{pmatrix}x_{i}^{5}&x_{i}^{10}&x_{i}^{6}&x_{i}^{11}\\ x_{i}^{8}&x_{i}^{1}&x_{i}^{9}&x_{k}^{11}\end{pmatrix}$ for $(i,k)=(1,12),(2,4),(5,15),(6,7),(7,13),(8,15),(10,1),(11,2),\\ (12,16),(14,15)$;

$\begin{pmatrix}x_{i}^{5}&x_{i}^{10}&x_{i}^{6}&x_{i}^{11}\\ x_{i}^{8}&x_{k_{1}}^{10}&x_{i}^{9}&x_{k_{2}}^{11}\end{pmatrix}$ for $(i,k_{1},k_{2})=(3,11,12),(4,10,11),(9,1,11),(13,6,5)$;

$\begin{pmatrix}x_{i}^{1}&x_{i}^{8}&x_{i}^{2}&x_{i}^{9}\\ x_{k}^{1}&x_{i}^{6}&x_{i}^{10}&x_{i}^{7}\end{pmatrix}$ for $(i,k)=(3,11),(4,10),(7,5),(8,6),(9,1),(10,2),(11,1),(13,6)$;

$\begin{pmatrix}x_{i}^{3}&x_{i}^{7}&x_{i}^{4}&x_{i}^{10}\\ x_{k}^{3}&x_{i}^{6}&x_{i}^{8}&x_{i}^{9}\end{pmatrix}$ for $(i,k)=(3,10),(4,1),(9,2),(13,7)$;

$\begin{pmatrix}x_{i}^{2}&x_{i}^{5}&x_{i}^{7}&x_{i}^{11}\\ x_{i}^{3}&x_{i}^{9}&x_{i}^{8}&x_{k}^{11}\end{pmatrix}$ for $(i,k)=(3,11),(4,10),(9,1),(13,6)$;

$\begin{pmatrix}x_{i}^{1}&x_{i}^{5}&x_{i}^{4}&x_{i}^{11}\\ x_{i}^{2}&x_{i}^{6}&x_{k}^{4}&x_{i}^{9}\end{pmatrix}$ for $(i,k)=(1,15),(2,16),(3,11),(4,10),(5,16),(6,15),(7,15),\\ (8,16),(9,1),(10,12),(11,16),(12,4),(13,6),(14,8)$;

$\begin{pmatrix}x_{5}^{3}&x_{6}^{3}&x_{11}^{3}&x_{12}^{3}\\ x_{15}^{3}&x_{14}^{3}&x_{2}^{3}&x_{16}^{3}\end{pmatrix}$, $\begin{pmatrix}x_{7}^{3}&x_{8}^{3}&x_{9}^{3}&x_{10}^{3}\\ x_{6}^{3}&x_{15}^{3}&x_{11}^{3}&x_{1}^{3}\end{pmatrix}$, $\begin{pmatrix}x_{1}^{1}&x_{13}^{1}&x_{2}^{1}&x_{14}^{1}\\ x_{15}^{1}&x_{13}^{10}&x_{16}^{1}&x_{5}^{1}\end{pmatrix}$, $\begin{pmatrix}x_{1}^{3}&x_{13}^{3}&x_{2}^{3}&x_{14}^{3}\\ x_{11}^{3}&x_{6}^{3}&x_{16}^{3}&x_{5}^{3}\end{pmatrix}$,

$\begin{pmatrix}x_{3}^{3}&x_{15}^{3}&x_{4}^{3}&x_{16}^{3}\\ x_{11}^{3}&x_{15}^{6}&x_{10}^{3}&x_{16}^{6}\end{pmatrix}$, $\begin{pmatrix}x_{7}^{3}&x_{11}^{3}&x_{8}^{3}&x_{12}^{3}\\ x_{9}^{3}&x_{15}^{3}&x_{5}^{3}&x_{6}^{3}\end{pmatrix}$, $\begin{pmatrix}x_{5}^{1}&x_{9}^{1}&x_{6}^{1}&x_{10}^{1}\\ x_{16}^{1}&x_{12}^{1}&x_{15}^{1}&x_{11}^{1}\end{pmatrix}$, $\begin{pmatrix}x_{3}^{3}&x_{5}^{3}&x_{4}^{3}&x_{6}^{3}\\ x_{2}^{3}&x_{11}^{3}&x_{9}^{3}&x_{16}^{3}\end{pmatrix}$,

$\begin{pmatrix}x_{1}^{3}&x_{7}^{3}&x_{2}^{3}&x_{8}^{3}\\ x_{15}^{3}&x_{16}^{3}&x_{10}^{3}&x_{13}^{3}\end{pmatrix}$, $\begin{pmatrix}x_{9}^{2}&x_{15}^{2}&x_{10}^{2}&x_{16}^{2}\\ x_{1}^{2}&x_{15}^{3}&x_{3}^{2}&x_{16}^{3}\end{pmatrix}$, $\begin{pmatrix}x_{9}^{3}&x_{15}^{3}&x_{10}^{3}&x_{16}^{3}\\ x_{1}^{3}&x_{12}^{3}&x_{8}^{3}&x_{14}^{3}\end{pmatrix}$, $\begin{pmatrix}x_{11}^{4}&x_{13}^{4}&x_{12}^{4}&x_{14}^{4}\\ x_{11}^{1}&x_{15}^{4}&x_{12}^{2}&x_{14}^{2}\end{pmatrix}$,

$\begin{pmatrix}x_{3}^{4}&x_{7}^{4}&x_{4}^{4}&x_{8}^{4}\\ x_{13}^{4}&x_{7}^{1}&x_{14}^{4}&x_{8}^{1}\end{pmatrix}$, $\begin{pmatrix}x_{1}^{3}&x_{5}^{3}&x_{2}^{3}&x_{6}^{3}\\ x_{3}^{3}&x_{16}^{3}&x_{12}^{3}&x_{15}^{3}\end{pmatrix}$, $\begin{pmatrix}x_{9}^{4}&x_{13}^{4}&x_{10}^{4}&x_{14}^{4}\\ x_{3}^{4}&x_{16}^{4}&x_{10}^{1}&x_{7}^{4}\end{pmatrix}$, $\begin{pmatrix}x_{9}^{5}&x_{15}^{5}&x_{10}^{5}&x_{16}^{5}\\ x_{1}^{5}&x_{15}^{8}&x_{3}^{5}&x_{16}^{8}\end{pmatrix}$.

4.2.3 An $L_{8}$- decomposition of $K_{16}\Box K_{13}$

$\begin{pmatrix}x_{1}^{j}&x_{2}^{j}&x_{15}^{j}&x_{16}^{j}\\ x_{12}^{j}&x_{4}^{j}&x_{6}^{j}&x_{5}^{j}\end{pmatrix}$, $\begin{pmatrix}x_{3}^{j}&x_{5}^{j}&x_{4}^{j}&x_{6}^{j}\\ x_{2}^{j}&x_{11}^{j}&x_{9}^{j}&x_{14}^{j}\end{pmatrix}$, for $j=2,\dots,8,10,\dots,13$;

$\begin{pmatrix}x_{5}^{j}&x_{6}^{j}&x_{11}^{j}&x_{12}^{j}\\ x_{15}^{j}&x_{7}^{j}&x_{2}^{j}&x_{16}^{j}\end{pmatrix}$, $\begin{pmatrix}x_{7}^{j}&x_{8}^{j}&x_{9}^{j}&x_{10}^{j}\\ x_{13}^{j}&x_{15}^{j}&x_{11}^{j}&x_{1}^{j}\end{pmatrix}$ for $j=1,\dots,13$;

$\begin{pmatrix}x_{1}^{j}&x_{7}^{j}&x_{2}^{j}&x_{8}^{j}\\ x_{4}^{j}&x_{16}^{j}&x_{9}^{j}&x_{13}^{j}\end{pmatrix}$, $\begin{pmatrix}x_{5}^{j}&x_{9}^{j}&x_{6}^{j}&x_{10}^{j}\\ x_{7}^{j}&x_{12}^{j}&x_{8}^{j}&x_{11}^{j}\end{pmatrix}$ for $j=1,\dots,13$;

$\begin{pmatrix}x_{1}^{j}&x_{13}^{j}&x_{2}^{j}&x_{14}^{j}\\ x_{11}^{j}&x_{6}^{j}&x_{10}^{j}&x_{5}^{j}\end{pmatrix}$, $\begin{pmatrix}x_{3}^{j}&x_{15}^{j}&x_{4}^{j}&x_{16}^{j}\\ x_{11}^{j}&x_{1}^{j}&x_{10}^{j}&x_{2}^{j}\end{pmatrix}$ for $j=2,\dots,6,8,10,12,13$;

$\begin{pmatrix}x_{9}^{j}&x_{15}^{j}&x_{10}^{j}&x_{16}^{j}\\ x_{1}^{j}&x_{12}^{j}&x_{3}^{j}&x_{14}^{j}\end{pmatrix}$, $\begin{pmatrix}x_{11}^{j}&x_{13}^{j}&x_{12}^{j}&x_{14}^{j}\\ x_{16}^{j}&x_{15}^{j}&x_{4}^{j}&x_{8}^{j}\end{pmatrix}$ for $j=2,\dots 6,8,10,12,13$;

$\begin{pmatrix}x_{7}^{j}&x_{11}^{j}&x_{8}^{j}&x_{12}^{j}\\ x_{9}^{j}&x_{15}^{j}&x_{10}^{j}&x_{6}^{j}\end{pmatrix}$, $\begin{pmatrix}x_{1}^{j}&x_{5}^{j}&x_{2}^{j}&x_{6}^{j}\\ x_{3}^{j}&x_{8}^{j}&x_{12}^{j}&x_{16}^{j}\end{pmatrix}$ for $j=1,\dots,6,8,9,10,12,13$;

, $\begin{pmatrix}x_{i}^{1}&x_{i}^{7}&x_{i}^{2}&x_{i}^{8}\\ x_{i}^{5}&x_{i}^{4}&x_{i}^{6}&x_{i}^{3}\end{pmatrix}$, $\begin{pmatrix}x_{i}^{4}&x_{i}^{5}&x_{i}^{7}&x_{i}^{6}\\ x_{i}^{2}&x_{i}^{3}&x_{i}^{10}&x_{i}^{1}\end{pmatrix}$ for $i=1,2,...,16$;

$\begin{pmatrix}x_{i}^{1}&x_{i}^{2}&x_{i}^{10}&x_{i}^{9}\\ x_{i}^{4}&x_{i}^{12}&x_{i}^{6}&x_{i}^{5}\end{pmatrix}$, $\begin{pmatrix}x_{i}^{5}&x_{i}^{6}&x_{i}^{13}&x_{i}^{10}\\ x_{i}^{2}&x_{i}^{3}&x_{i}^{11}&x_{i}^{12}\end{pmatrix}$, for $i=1,2,...,16$;

$\begin{pmatrix}x_{7}^{j}&x_{11}^{j}&x_{8}^{j}&x_{12}^{j}\\ x_{14}^{j}&x_{15}^{j}&x_{10}^{j}&x_{6}^{j}\end{pmatrix}$, $\begin{pmatrix}x_{1}^{j}&x_{13}^{j}&x_{2}^{j}&x_{14}^{j}\\ x_{11}^{j}&x_{6}^{j}&x_{16}^{j}&x_{5}^{j}\end{pmatrix}$, for $j=7,11$;

$\begin{pmatrix}x_{i}^{3}&x_{i}^{4}&x_{i}^{8}&x_{i}^{7}\\ x_{i}^{13}&x_{i}^{11}&x_{i}^{5}&x_{i}^{9}\end{pmatrix}$, $\begin{pmatrix}x_{i}^{2}&x_{i}^{3}&x_{i}^{12}&x_{i}^{11}\\ x_{i}^{13}&x_{i}^{1}&x_{i}^{8}&x_{i}^{10}\end{pmatrix}$, for $i=1,2,5,6,7,8,11,\dots,16$;

$\begin{pmatrix}x_{3}^{j}&x_{4}^{j}&x_{13}^{j}&x_{14}^{j}\\ x_{3}^{k_{1}}&x_{4}^{k_{1}}&x_{13}^{k_{2}}&x_{15}^{j}\end{pmatrix}$, $\begin{pmatrix}x_{9}^{j}&x_{13}^{j}&x_{10}^{j}&x_{14}^{j}\\ x_{9}^{k_{1}}&x_{16}^{j}&x_{10}^{k_{1}}&x_{14}^{k_{2}}\end{pmatrix}$ for $(j,k_{1},k_{2})=(1,10,12),(9,7,13)$;

$\begin{pmatrix}x_{3}^{j}&x_{15}^{j}&x_{4}^{j}&x_{16}^{j}\\ x_{11}^{j}&x_{13}^{j}&x_{10}^{j}&x_{16}^{k}\end{pmatrix}$, $\begin{pmatrix}x_{11}^{j}&x_{13}^{j}&x_{12}^{j}&x_{14}^{j}\\ x_{16}^{j}&x_{13}^{k}&x_{15}^{j}&x_{8}^{j}\end{pmatrix}$ for $(j,k)=(7,13),(11,8)$;

$\begin{pmatrix}x_{i}^{3}&x_{i}^{4}&x_{i}^{8}&x_{i}^{7}\\ x_{i}^{13}&x_{i}^{11}&x_{i}^{5}&x_{k}^{7}\end{pmatrix}$, $\begin{pmatrix}x_{i}^{2}&x_{i}^{3}&x_{i}^{12}&x_{i}^{11}\\ x_{i}^{13}&x_{i}^{1}&x_{i}^{8}&x_{k}^{11}\end{pmatrix}$ for $(i,k)=(3,1),(4,12),(9,7),(10,2)$;

$\begin{pmatrix}x_{3}^{j}&x_{5}^{j}&x_{4}^{j}&x_{6}^{j}\\ x_{2}^{j}&x_{13}^{j}&x_{9}^{j}&x_{14}^{j}\end{pmatrix}$ for $j=1,9$; $\begin{pmatrix}x_{1}^{j}&x_{2}^{j}&x_{15}^{j}&x_{16}^{j}\\ x_{12}^{j}&x_{4}^{j}&x_{6}^{j}&x_{16}^{k}\end{pmatrix}$ for $(j,k)=(1,12),(9,11)$;

$\begin{pmatrix}x_{3}^{j}&x_{7}^{j}&x_{4}^{j}&x_{8}^{j}\\ x_{13}^{j}&x_{15}^{j}&x_{14}^{j}&x_{16}^{j}\end{pmatrix}$ for $j=1,\dots,13$; $\begin{pmatrix}x_{11}^{j}&x_{13}^{j}&x_{12}^{j}&x_{14}^{j}\\ x_{16}^{j}&x_{15}^{j}&x_{4}^{j}&x_{14}^{k}\end{pmatrix}$ for $(j,k)=(1,11),(9,11)$;

$\begin{pmatrix}x_{1}^{j}&x_{5}^{j}&x_{2}^{j}&x_{6}^{j}\\ x_{15}^{j}&x_{8}^{j}&x_{12}^{j}&x_{16}^{j}\end{pmatrix}$ for $j=7,11$; $\begin{pmatrix}x_{9}^{j}&x_{13}^{j}&x_{10}^{j}&x_{14}^{j}\\ x_{3}^{j}&x_{16}^{j}&x_{12}^{j}&x_{7}^{j}\end{pmatrix}$ for $j=2,\dots,6,8,12,13$;

$\begin{pmatrix}x_{3}^{j}&x_{4}^{j}&x_{13}^{j}&x_{14}^{j}\\ x_{12}^{j}&x_{11}^{j}&x_{5}^{j}&x_{15}^{j}\end{pmatrix}$ for $j=2,\dots,8,11,12,13$;

$\begin{pmatrix}x_{9}^{j}&x_{13}^{j}&x_{10}^{j}&x_{14}^{j}\\ x_{3}^{j}&x_{16}^{j}&x_{12}^{j}&x_{14}^{k}\end{pmatrix}$ for $(j,k)=(7,13),(11,8)$;

$\begin{pmatrix}x_{9}^{j}&x_{15}^{j}&x_{10}^{j}&x_{16}^{j}\\ x_{1}^{j}&x_{15}^{k}&x_{3}^{j}&x_{14}^{j}\end{pmatrix}$ for $(j,k)=(1,12),(7,13),(9,13),(11,8)$;

$\begin{pmatrix}x_{i}^{3}&x_{i}^{9}&x_{i}^{4}&x_{i}^{10}\\ x_{i}^{11}&x_{i}^{2}&x_{i}^{k}&x_{i}^{1}\end{pmatrix}$ for $i=1,2,5,6,7,8,11,12,\ k=12\ \&\ i=13,14,15,16\ k=13$;

$\begin{pmatrix}x_{i}^{3}&x_{i}^{9}&x_{i}^{4}&x_{i}^{10}\\ x_{i}^{11}&x_{i}^{2}&x_{i}^{12}&x_{k}^{10}\end{pmatrix}$ for $(i,k)=(3,12),(4,11),(9,3),(10,12)$;

$\begin{pmatrix}x_{i}^{5}&x_{i}^{11}&x_{i}^{6}&x_{i}^{12}\\ x_{i}^{13}&x_{i}^{7}&x_{i}^{8}&x_{i}^{k}\end{pmatrix}$ for $i=1,2,...,12,\ k=1\ \&\ i=13,14,15,16\ k=4$;

$\begin{pmatrix}x_{i}^{8}&x_{i}^{9}&x_{i}^{12}&x_{i}^{13}\\ x_{i}^{10}&x_{i}^{6}&x_{i}^{7}&x_{i}^{k}\end{pmatrix}$ for $i=1,2,...,12,\ k=4\ \&\ i=13,14,15,16\ k=1$;

$\begin{pmatrix}x_{i}^{1}&x_{i}^{11}&x_{i}^{9}&x_{i}^{13}\\ x_{k}^{1}&x_{i}^{8}&x_{k}^{9}&x_{i}^{7}\end{pmatrix}$ for $(i,k)=(1,15),(2,16),(3,12),(4,11),(5,16),(6,13),(7,14)$,

$(8,14),(9,3),(10,12),(11,5),(12,15)$;

$\begin{pmatrix}x_{3}^{10}&x_{4}^{10}&x_{13}^{10}&x_{14}^{10}\\ x_{3}^{11}&x_{4}^{11}&x_{5}^{10}&x_{15}^{10}\end{pmatrix}$, $\begin{pmatrix}x_{1}^{1}&x_{13}^{1}&x_{2}^{1}&x_{14}^{1}\\ x_{11}^{1}&x_{13}^{11}&x_{10}^{1}&x_{5}^{1}\end{pmatrix}$, $\begin{pmatrix}x_{1}^{9}&x_{13}^{9}&x_{2}^{9}&x_{14}^{9}\\ x_{11}^{9}&x_{13}^{11}&x_{10}^{9}&x_{5}^{9}\end{pmatrix}$, $\begin{pmatrix}x_{3}^{1}&x_{15}^{1}&x_{4}^{1}&x_{16}^{1}\\ x_{11}^{1}&x_{15}^{11}&x_{10}^{1}&x_{16}^{11}\end{pmatrix}$,

$\begin{pmatrix}x_{3}^{9}&x_{15}^{9}&x_{4}^{9}&x_{16}^{9}\\ x_{11}^{9}&x_{15}^{11}&x_{10}^{9}&x_{16}^{13}\end{pmatrix}$, $\begin{pmatrix}x_{9}^{10}&x_{13}^{10}&x_{10}^{10}&x_{14}^{10}\\ x_{9}^{11}&x_{16}^{10}&x_{10}^{11}&x_{7}^{10}\end{pmatrix}$.

4.2.4 An $L_{8}$- decomposition of $K_{16}\Box K_{7}$

$\begin{pmatrix}x_{1}^{j}&x_{2}^{j}&x_{15}^{j}&x_{16}^{j}\\ x_{1}^{k}&x_{2}^{k}&x_{15}^{k}&x_{16}^{k}\end{pmatrix}$, $\begin{pmatrix}x_{3}^{j}&x_{4}^{j}&x_{13}^{j}&x_{14}^{j}\\ x_{3}^{k}&x_{4}^{k}&x_{13}^{k}&x_{14}^{k}\end{pmatrix}$ for $(j,k)=(1,4),(2,3),(3,6),(4,3),(5,6)$,

$(6,2),(7,6)$;

$\begin{pmatrix}x_{1}^{j}&x_{13}^{j}&x_{2}^{j}&x_{14}^{j}\\ x_{11}^{j}&x_{6}^{j}&x_{10}^{j}&x_{5}^{j}\end{pmatrix}$, $\begin{pmatrix}x_{1}^{j}&x_{7}^{j}&x_{2}^{j}&x_{8}^{j}\\ x_{4}^{j}&x_{16}^{j}&x_{9}^{j}&x_{13}^{j}\end{pmatrix}$ for $j=1,3,6,7$;

$\begin{pmatrix}x_{1}^{j}&x_{7}^{j}&x_{2}^{j}&x_{8}^{j}\\ x_{4}^{j}&x_{16}^{j}&x_{11}^{j}&x_{13}^{j}\end{pmatrix}$ for $j=4,5$; $\begin{pmatrix}x_{7}^{j}&x_{11}^{j}&x_{8}^{j}&x_{12}^{j}\\ x_{9}^{j}&x_{15}^{j}&x_{10}^{j}&x_{6}^{j}\end{pmatrix}$ for $j=1,3,4,6$;

$\begin{pmatrix}x_{9}^{j}&x_{15}^{j}&x_{10}^{j}&x_{16}^{j}\\ x_{1}^{j}&x_{12}^{j}&x_{3}^{j}&x_{14}^{j}\end{pmatrix}$ for $j=1,6$; $\begin{pmatrix}x_{5}^{j}&x_{9}^{j}&x_{6}^{j}&x_{10}^{j}\\ x_{7}^{j}&x_{12}^{j}&x_{8}^{j}&x_{11}^{j}\end{pmatrix}$ for $j=1,4,6,7$;

$\begin{pmatrix}x_{11}^{j}&x_{13}^{j}&x_{12}^{j}&x_{14}^{j}\\ x_{16}^{j}&x_{15}^{j}&x_{4}^{j}&x_{8}^{j}\end{pmatrix}$ for $j=1,6$; $\begin{pmatrix}x_{3}^{j}&x_{7}^{j}&x_{4}^{j}&x_{8}^{j}\\ x_{13}^{j}&x_{15}^{j}&x_{14}^{j}&x_{16}^{j}\end{pmatrix}$ for $j=1,2,4,5,6$;

$\begin{pmatrix}x_{3}^{j}&x_{7}^{j}&x_{4}^{j}&x_{8}^{j}\\ x_{10}^{j}&x_{15}^{j}&x_{14}^{j}&x_{16}^{j}\end{pmatrix}$ for $j=3,7$; $\begin{pmatrix}x_{1}^{j}&x_{5}^{j}&x_{2}^{j}&x_{6}^{j}\\ x_{3}^{j}&x_{8}^{j}&x_{12}^{j}&x_{16}^{j}\end{pmatrix}$ for $j=1,3,4,5,6,7$;

$\begin{pmatrix}x_{3}^{j}&x_{15}^{j}&x_{4}^{j}&x_{16}^{j}\\ x_{11}^{j}&x_{1}^{j}&x_{10}^{j}&x_{2}^{j}\end{pmatrix}$ for $j=1,6,7$; $\begin{pmatrix}x_{1}^{j}&x_{13}^{j}&x_{2}^{j}&x_{14}^{j}\\ x_{11}^{j}&x_{13}^{k}&x_{10}^{j}&x_{5}^{j}\end{pmatrix}$ for $(j,k)=(2,5),(4,7),(5,4)$;

$\begin{pmatrix}x_{3}^{j}&x_{5}^{j}&x_{4}^{j}&x_{6}^{j}\\ x_{2}^{j}&x_{11}^{j}&x_{9}^{j}&x_{14}^{j}\end{pmatrix}$ for $j=1,...,7$; $\begin{pmatrix}x_{3}^{j}&x_{15}^{j}&x_{4}^{j}&x_{16}^{j}\\ x_{11}^{j}&x_{15}^{k}&x_{10}^{j}&x_{16}^{k}\end{pmatrix}$ $(j,k)=(2,5),(4,7),(5,4)$;

$\begin{pmatrix}x_{9}^{j}&x_{13}^{j}&x_{10}^{j}&x_{14}^{j}\\ x_{3}^{j}&x_{16}^{j}&x_{12}^{j}&x_{7}^{j}\end{pmatrix}$ for $j=1,3,4,5,6,7$;

$\begin{pmatrix}x_{7}^{j}&x_{8}^{j}&x_{9}^{j}&x_{10}^{j}\\ x_{7}^{k}&x_{8}^{k}&x_{9}^{k}&x_{10}^{k}\end{pmatrix}$ for $(j,k)=(1,4),(5,6),(6,2),(7,6)$;

$\begin{pmatrix}x_{7}^{j}&x_{8}^{j}&x_{9}^{j}&x_{10}^{j}\\ x_{7}^{k_{1}}&x_{8}^{k_{2}}&x_{9}^{k_{2}}&x_{10}^{k_{2}}\end{pmatrix}$ for $(j,k_{1},k_{2})=(2,4,3),(3,4,6),(4,6,3)$;

$\begin{pmatrix}x_{9}^{j}&x_{15}^{j}&x_{10}^{j}&x_{16}^{j}\\ x_{1}^{j}&x_{12}^{j}&x_{10}^{k}&x_{14}^{j}\end{pmatrix}$ for $(j,k)=(3,7),(4,7),(5,2),(7,2)$;

$\begin{pmatrix}x_{5}^{j}&x_{6}^{j}&x_{11}^{j}&x_{12}^{j}\\ x_{5}^{k}&x_{6}^{k}&x_{11}^{k}&x_{12}^{k}\end{pmatrix}$ for $(j,k)=(1,4),(4,3),(5,6),(7,6)$;

$\begin{pmatrix}x_{i}^{1}&x_{i}^{3}&x_{i}^{5}&x_{i}^{7}\\ x_{k}^{1}&x_{k}^{3}&x_{k}^{5}&x_{k}^{7}\end{pmatrix}$ for $(i,k)=(1,12),(2,4),(3,12),(4,11),(8,15),(9,11),(10,1)$,

$(12,16),(13,5)$;

$\begin{pmatrix}x_{i}^{1}&x_{i}^{2}&x_{i}^{4}&x_{i}^{6}\\ x_{i}^{5}&x_{k}^{2}&x_{k}^{4}&x_{k}^{6}\end{pmatrix}$ for $(i,k)=(1,12),(2,4),(3,12),(4,11),(5,15),(8,15),(12,16)$,

$(14,15)$;

$\begin{pmatrix}x_{i}^{2}&x_{i}^{5}&x_{i}^{4}&x_{i}^{7}\\ x_{k}^{2}&x_{k}^{5}&x_{k}^{4}&x_{i}^{3}\end{pmatrix}$ for $(i,k)=(1,15),(2,16),(4,12),(5,16),(6,13),(8,14),(9,2)$;

$\begin{pmatrix}x_{5}^{2}&x_{6}^{2}&x_{11}^{2}&x_{12}^{2}\\ x_{7}^{2}&x_{6}^{3}&x_{11}^{3}&x_{12}^{3}\end{pmatrix}$, $\begin{pmatrix}x_{5}^{3}&x_{6}^{3}&x_{11}^{3}&x_{12}^{3}\\ x_{5}^{6}&x_{6}^{6}&x_{11}^{6}&x_{12}^{7}\end{pmatrix}$, $\begin{pmatrix}x_{5}^{6}&x_{6}^{6}&x_{11}^{6}&x_{12}^{6}\\ x_{5}^{2}&x_{6}^{2}&x_{11}^{2}&x_{12}^{3}\end{pmatrix}$, $\begin{pmatrix}x_{3}^{3}&x_{15}^{3}&x_{4}^{3}&x_{16}^{3}\\ x_{11}^{3}&x_{1}^{3}&x_{10}^{3}&x_{16}^{7}\end{pmatrix}$,

$\begin{pmatrix}x_{7}^{2}&x_{11}^{2}&x_{8}^{2}&x_{12}^{2}\\ x_{16}^{2}&x_{11}^{7}&x_{6}^{2}&x_{12}^{7}\end{pmatrix}$, $\begin{pmatrix}x_{7}^{5}&x_{11}^{5}&x_{8}^{5}&x_{12}^{5}\\ x_{9}^{5}&x_{15}^{5}&x_{10}^{5}&x_{12}^{2}\end{pmatrix}$, $\begin{pmatrix}x_{7}^{7}&x_{11}^{7}&x_{8}^{7}&x_{12}^{7}\\ x_{9}^{7}&x_{11}^{3}&x_{10}^{7}&x_{6}^{7}\end{pmatrix}$, $\begin{pmatrix}x_{5}^{2}&x_{9}^{2}&x_{6}^{2}&x_{10}^{2}\\ x_{13}^{2}&x_{11}^{2}&x_{12}^{2}&x_{1}^{2}\end{pmatrix}$,

$\begin{pmatrix}x_{5}^{3}&x_{9}^{3}&x_{6}^{3}&x_{10}^{3}\\ x_{16}^{3}&x_{12}^{3}&x_{8}^{3}&x_{11}^{3}\end{pmatrix}$, $\begin{pmatrix}x_{5}^{5}&x_{9}^{5}&x_{6}^{5}&x_{10}^{5}\\ x_{7}^{5}&x_{12}^{5}&x_{8}^{5}&x_{10}^{4}\end{pmatrix}$, $\begin{pmatrix}x_{1}^{2}&x_{7}^{2}&x_{2}^{2}&x_{8}^{2}\\ x_{4}^{2}&x_{6}^{2}&x_{11}^{2}&x_{10}^{2}\end{pmatrix}$, $\begin{pmatrix}x_{9}^{2}&x_{15}^{2}&x_{10}^{2}&x_{16}^{2}\\ x_{7}^{2}&x_{12}^{2}&x_{3}^{2}&x_{14}^{2}\end{pmatrix}$,

$\begin{pmatrix}x_{11}^{2}&x_{13}^{2}&x_{12}^{2}&x_{14}^{2}\\ x_{16}^{2}&x_{15}^{2}&x_{12}^{6}&x_{14}^{5}\end{pmatrix}$, $\begin{pmatrix}x_{11}^{3}&x_{13}^{3}&x_{12}^{3}&x_{14}^{3}\\ x_{16}^{3}&x_{7}^{3}&x_{4}^{3}&x_{14}^{7}\end{pmatrix}$, $\begin{pmatrix}x_{11}^{4}&x_{13}^{4}&x_{12}^{4}&x_{14}^{4}\\ x_{11}^{5}&x_{15}^{4}&x_{12}^{7}&x_{14}^{7}\end{pmatrix}$, $\begin{pmatrix}x_{11}^{5}&x_{13}^{5}&x_{12}^{5}&x_{14}^{5}\\ x_{10}^{5}&x_{15}^{5}&x_{12}^{4}&x_{14}^{4}\end{pmatrix}$,

$\begin{pmatrix}x_{11}^{7}&x_{13}^{7}&x_{12}^{7}&x_{14}^{7}\\ x_{16}^{7}&x_{13}^{2}&x_{4}^{7}&x_{8}^{7}\end{pmatrix}$, $\begin{pmatrix}x_{1}^{2}&x_{5}^{2}&x_{2}^{2}&x_{6}^{2}\\ x_{9}^{2}&x_{8}^{2}&x_{12}^{2}&x_{16}^{2}\end{pmatrix}$, $\begin{pmatrix}x_{9}^{2}&x_{13}^{2}&x_{10}^{2}&x_{14}^{2}\\ x_{3}^{2}&x_{16}^{2}&x_{12}^{2}&x_{14}^{7}\end{pmatrix}$, $\begin{pmatrix}x_{5}^{1}&x_{5}^{3}&x_{5}^{5}&x_{5}^{7}\\ x_{15}^{1}&x_{5}^{2}&x_{15}^{5}&x_{15}^{7}\end{pmatrix}$,

$\begin{pmatrix}x_{6}^{1}&x_{6}^{3}&x_{6}^{5}&x_{6}^{7}\\ x_{7}^{1}&x_{15}^{3}&x_{12}^{5}&x_{7}^{7}\end{pmatrix}$, $\begin{pmatrix}x_{7}^{1}&x_{7}^{3}&x_{7}^{5}&x_{7}^{7}\\ x_{13}^{1}&x_{5}^{3}&x_{13}^{5}&x_{13}^{7}\end{pmatrix}$, $\begin{pmatrix}x_{11}^{1}&x_{11}^{3}&x_{11}^{5}&x_{11}^{7}\\ x_{2}^{1}&x_{2}^{3}&x_{11}^{2}&x_{2}^{7}\end{pmatrix}$, $\begin{pmatrix}x_{14}^{1}&x_{14}^{3}&x_{14}^{5}&x_{14}^{7}\\ x_{15}^{1}&x_{8}^{3}&x_{15}^{5}&x_{15}^{7}\end{pmatrix}$,

$\begin{pmatrix}x_{15}^{1}&x_{15}^{3}&x_{15}^{5}&x_{15}^{7}\\ x_{6}^{1}&x_{5}^{3}&x_{6}^{5}&x_{6}^{7}\end{pmatrix}$, $\begin{pmatrix}x_{16}^{1}&x_{16}^{3}&x_{16}^{5}&x_{16}^{7}\\ x_{5}^{1}&x_{2}^{3}&x_{11}^{5}&x_{5}^{7}\end{pmatrix}$, $\begin{pmatrix}x_{6}^{1}&x_{6}^{2}&x_{6}^{4}&x_{6}^{6}\\ x_{6}^{5}&x_{15}^{2}&x_{7}^{4}&x_{7}^{6}\end{pmatrix}$, $\begin{pmatrix}x_{9}^{1}&x_{9}^{2}&x_{9}^{4}&x_{9}^{6}\\ x_{9}^{5}&x_{12}^{2}&x_{11}^{4}&x_{11}^{6}\end{pmatrix}$,

$\begin{pmatrix}x_{10}^{1}&x_{10}^{2}&x_{10}^{4}&x_{10}^{6}\\ x_{10}^{5}&x_{11}^{2}&x_{1}^{4}&x_{1}^{6}\end{pmatrix}$, $\begin{pmatrix}x_{11}^{1}&x_{11}^{2}&x_{11}^{4}&x_{11}^{6}\\ x_{11}^{5}&x_{15}^{2}&x_{11}^{7}&x_{2}^{6}\end{pmatrix}$, $\begin{pmatrix}x_{13}^{1}&x_{13}^{2}&x_{13}^{4}&x_{13}^{6}\\ x_{13}^{5}&x_{8}^{2}&x_{5}^{4}&x_{5}^{6}\end{pmatrix}$, $\begin{pmatrix}x_{15}^{1}&x_{15}^{2}&x_{15}^{4}&x_{15}^{6}\\ x_{15}^{5}&x_{15}^{7}&x_{6}^{4}&x_{6}^{6}\end{pmatrix}$,

$\begin{pmatrix}x_{16}^{1}&x_{16}^{2}&x_{16}^{4}&x_{16}^{6}\\ x_{16}^{5}&x_{16}^{7}&x_{11}^{4}&x_{5}^{6}\end{pmatrix}$, $\begin{pmatrix}x_{3}^{2}&x_{3}^{5}&x_{3}^{4}&x_{3}^{7}\\ x_{1}^{2}&x_{10}^{5}&x_{10}^{4}&x_{3}^{3}\end{pmatrix}$, $\begin{pmatrix}x_{7}^{2}&x_{7}^{5}&x_{7}^{4}&x_{7}^{7}\\ x_{14}^{2}&x_{6}^{5}&x_{13}^{4}&x_{7}^{3}\end{pmatrix}$, $\begin{pmatrix}x_{7}^{1}&x_{7}^{2}&x_{7}^{3}&x_{7}^{6}\\ x_{7}^{5}&x_{13}^{2}&x_{6}^{3}&x_{13}^{6}\end{pmatrix}$,

$\begin{pmatrix}x_{13}^{3}&x_{13}^{7}&x_{15}^{7}&x_{15}^{3}\\ x_{3}^{3}&x_{3}^{7}&x_{11}^{7}&x_{14}^{3}\end{pmatrix}$.

4.3 $L_{8}$-decomposition required for Lemma 2.6

4.3.1 An $L_{8}$- decomposition of $K_{19}\backslash K_{3}$

Let $V(K_{19})=\left\{x_{1},x_{2},...,x_{19}\right\}$ and the $K_{3}$ be $(x_{13}x_{15}x_{17})$.

$\begin{pmatrix}x_{1}&x_{2}&x_{18}&x_{19}\\ x_{9}&x_{12}&x_{5}&x_{6}\end{pmatrix}$, $\begin{pmatrix}x_{3}&x_{4}&x_{16}&x_{17}\\ x_{9}&x_{1}&x_{18}&x_{19}\end{pmatrix}$, $\begin{pmatrix}x_{5}&x_{6}&x_{14}&x_{15}\\ x_{3}&x_{4}&x_{16}&x_{1}\end{pmatrix}$, $\begin{pmatrix}x_{7}&x_{8}&x_{12}&x_{13}\\ x_{9}&x_{2}&x_{16}&x_{14}\end{pmatrix}$, $\begin{pmatrix}x_{9}&x_{10}&x_{11}&x_{12}\\ x_{4}&x_{16}&x_{7}&x_{17}\end{pmatrix}$, $\begin{pmatrix}x_{1}&x_{16}&x_{2}&x_{17}\\ x_{12}&x_{3}&x_{9}&x_{4}\end{pmatrix}$, $\begin{pmatrix}x_{3}&x_{18}&x_{4}&x_{19}\\ x_{10}&x_{1}&x_{2}&x_{12}\end{pmatrix}$, $\begin{pmatrix}x_{5}&x_{12}&x_{6}&x_{13}\\ x_{11}&x_{18}&x_{3}&x_{16}\end{pmatrix}$, $\begin{pmatrix}x_{7}&x_{14}&x_{8}&x_{15}\\ x_{2}&x_{17}&x_{1}&x_{3}\end{pmatrix}$, $\begin{pmatrix}x_{1}&x_{10}&x_{2}&x_{11}\\ x_{14}&x_{13}&x_{19}&x_{3}\end{pmatrix}$, $\begin{pmatrix}x_{3}&x_{12}&x_{4}&x_{13}\\ x_{1}&x_{7}&x_{11}&x_{2}\end{pmatrix}$, $\begin{pmatrix}x_{1}&x_{5}&x_{2}&x_{6}\\ x_{13}&x_{14}&x_{15}&x_{11}\end{pmatrix}$,

$\begin{pmatrix}x_{7}&x_{16}&x_{8}&x_{17}\\ x_{6}&x_{19}&x_{5}&x_{11}\end{pmatrix}$, $\begin{pmatrix}x_{9}&x_{18}&x_{10}&x_{19}\\ x_{11}&x_{6}&x_{4}&x_{5}\end{pmatrix}$, $\begin{pmatrix}x_{11}&x_{14}&x_{12}&x_{15}\\ x_{13}&x_{2}&x_{10}&x_{4}\end{pmatrix}$, $\begin{pmatrix}x_{3}&x_{7}&x_{4}&x_{8}\\ x_{2}&x_{1}&x_{14}&x_{9}\end{pmatrix}$,

$\begin{pmatrix}x_{5}&x_{16}&x_{6}&x_{17}\\ x_{7}&x_{9}&x_{8}&x_{18}\end{pmatrix}$, $\begin{pmatrix}x_{9}&x_{14}&x_{10}&x_{15}\\ x_{13}&x_{3}&x_{17}&x_{16}\end{pmatrix}$, $\begin{pmatrix}x_{11}&x_{18}&x_{13}&x_{19}\\ x_{16}&x_{14}&x_{8}&x_{15}\end{pmatrix}$, $\begin{pmatrix}x_{5}&x_{9}&x_{6}&x_{10}\\ x_{4}&x_{17}&x_{15}&x_{8}\end{pmatrix}$,

$\begin{pmatrix}x_{7}&x_{18}&x_{8}&x_{19}\\ x_{10}&x_{15}&x_{11}&x_{14}\end{pmatrix}$.

4.3.2 An $L_{8}$- decomposition of $K_{16}\Box K_{3}$

$\begin{pmatrix}x_{1}^{j}&x_{2}^{j}&x_{15}^{j}&x_{16}^{j}\\ x_{12}^{j}&x_{4}^{j}&x_{6}^{j}&x_{5}^{j}\end{pmatrix}$, $\begin{pmatrix}x_{3}^{j}&x_{15}^{j}&x_{4}^{j}&x_{16}^{j}\\ x_{11}^{j}&x_{1}^{j}&x_{10}^{j}&x_{2}^{j}\end{pmatrix}$, $\begin{pmatrix}x_{11}^{j}&x_{13}^{j}&x_{12}^{j}&x_{14}^{j}\\ x_{16}^{j}&x_{15}^{j}&x_{4}^{j}&x_{8}^{j}\end{pmatrix}$ for $j=1,2,3$;

$\begin{pmatrix}x_{5}^{j}&x_{6}^{j}&x_{11}^{j}&x_{12}^{j}\\ x_{15}^{j}&x_{7}^{j}&x_{2}^{j}&x_{16}^{j}\end{pmatrix}$, $\begin{pmatrix}x_{1}^{j}&x_{13}^{j}&x_{2}^{j}&x_{14}^{j}\\ x_{11}^{j}&x_{6}^{j}&x_{10}^{j}&x_{5}^{j}\end{pmatrix}$ for $j=1,3$;

$\begin{pmatrix}x_{5}^{j}&x_{9}^{j}&x_{6}^{j}&x_{10}^{j}\\ x_{7}^{j}&x_{12}^{j}&x_{8}^{j}&x_{10}^{1}\end{pmatrix}$ for $j=2,3$;

$\begin{pmatrix}x_{7}^{j}&x_{8}^{j}&x_{9}^{j}&x_{10}^{j}\\ x_{13}^{j}&x_{15}^{j}&x_{k_{1}}^{k_{2}}&x_{1}^{j}\end{pmatrix}$ for $(j,k_{1},k_{2})=(1,9,3),(2,9,3),(3,11,3)$;

$\begin{pmatrix}x_{7}^{j}&x_{11}^{j}&x_{8}^{j}&x_{12}^{j}\\ x_{9}^{j}&x_{15}^{j}&x_{k_{1}}^{k_{2}}&x_{6}^{j}\end{pmatrix}$ for $(j,k_{1},k_{2})=(1,10,1),(2,10,2),(3,8,1)$;

$\begin{pmatrix}x_{3}^{j}&x_{5}^{j}&x_{4}^{j}&x_{6}^{j}\\ x_{2}^{j}&x_{k_{1}}^{k_{2}}&x_{9}^{j}&x_{14}^{j}\end{pmatrix}$ for $(j,k_{1},k_{2})=(1,11,1),(2,11,2),(3,5,2)$;

$\begin{pmatrix}x_{9}^{j}&x_{15}^{j}&x_{10}^{j}&x_{16}^{j}\\ x_{1}^{j}&x_{k_{1}}^{k_{2}}&x_{3}^{j}&x_{14}^{j}\end{pmatrix}$ for $(j,k_{1},k_{2})=(1,12,1),(2,15,3),(3,12,3)$;

$\begin{pmatrix}x_{3}^{j}&x_{7}^{j}&x_{4}^{j}&x_{8}^{j}\\ x_{13}^{j}&x_{15}^{j}&x_{k_{1}}^{k_{2}}&x_{16}^{j}\end{pmatrix}$ $(j,k_{1},k_{2})=(1,4,3),(2,4,1),(3,14,3)$;

$\begin{pmatrix}x_{9}^{j}&x_{13}^{j}&x_{10}^{j}&x_{14}^{j}\\ x_{3}^{j}&x_{k_{1}}^{k_{2}}&x_{12}^{j}&x_{7}^{j}\end{pmatrix}$ for $(j,k_{1},k_{2})=(1,16,1),(2,13,1),(3,13,1)$;

$\begin{pmatrix}x_{3}^{1}&x_{4}^{1}&x_{13}^{1}&x_{14}^{1}\\ x_{3}^{3}&x_{11}^{1}&x_{5}^{1}&x_{14}^{3}\end{pmatrix}$, $\begin{pmatrix}x_{3}^{2}&x_{4}^{2}&x_{13}^{2}&x_{14}^{2}\\ x_{3}^{3}&x_{4}^{3}&x_{5}^{2}&x_{14}^{3}\end{pmatrix}$, $\begin{pmatrix}x_{3}^{3}&x_{4}^{3}&x_{13}^{3}&x_{14}^{3}\\ x_{12}^{3}&x_{11}^{3}&x_{5}^{3}&x_{15}^{3}\end{pmatrix}$, $\begin{pmatrix}x_{5}^{2}&x_{6}^{2}&x_{11}^{2}&x_{12}^{2}\\ x_{5}^{1}&x_{6}^{1}&x_{2}^{2}&x_{12}^{3}\end{pmatrix}$,

$\begin{pmatrix}x_{1}^{2}&x_{13}^{2}&x_{2}^{2}&x_{14}^{2}\\ x_{1}^{1}&x_{6}^{2}&x_{2}^{1}&x_{5}^{2}\end{pmatrix}$, $\begin{pmatrix}x_{5}^{1}&x_{9}^{1}&x_{6}^{1}&x_{10}^{1}\\ x_{5}^{3}&x_{12}^{1}&x_{8}^{1}&x_{11}^{1}\end{pmatrix}$, $\begin{pmatrix}x_{1}^{1}&x_{7}^{1}&x_{2}^{1}&x_{8}^{1}\\ x_{4}^{1}&x_{7}^{3}&x_{2}^{3}&x_{8}^{2}\end{pmatrix}$, $\begin{pmatrix}x_{1}^{2}&x_{7}^{2}&x_{2}^{2}&x_{8}^{2}\\ x_{4}^{2}&x_{7}^{3}&x_{2}^{3}&x_{8}^{3}\end{pmatrix}$,

$\begin{pmatrix}x_{1}^{3}&x_{7}^{3}&x_{2}^{3}&x_{8}^{3}\\ x_{4}^{3}&x_{16}^{3}&x_{9}^{3}&x_{13}^{3}\end{pmatrix}$, $\begin{pmatrix}x_{1}^{1}&x_{5}^{1}&x_{2}^{1}&x_{6}^{1}\\ x_{1}^{3}&x_{8}^{1}&x_{12}^{1}&x_{16}^{1}\end{pmatrix}$, $\begin{pmatrix}x_{1}^{2}&x_{5}^{2}&x_{2}^{2}&x_{6}^{2}\\ x_{1}^{3}&x_{8}^{2}&x_{12}^{2}&x_{6}^{3}\end{pmatrix}$, $\begin{pmatrix}x_{1}^{3}&x_{5}^{3}&x_{2}^{3}&x_{6}^{3}\\ x_{3}^{3}&x_{8}^{3}&x_{12}^{3}&x_{6}^{1}\end{pmatrix}$,

$\begin{pmatrix}x_{3}^{1}&x_{12}^{1}&x_{12}^{2}&x_{3}^{2}\\ x_{1}^{1}&x_{12}^{3}&x_{15}^{2}&x_{1}^{2}\end{pmatrix}$, $\begin{pmatrix}x_{14}^{1}&x_{15}^{1}&x_{15}^{2}&x_{14}^{2}\\ x_{4}^{1}&x_{15}^{3}&x_{5}^{2}&x_{4}^{2}\end{pmatrix}$, $\begin{pmatrix}x_{9}^{1}&x_{11}^{1}&x_{11}^{2}&x_{9}^{2}\\ x_{2}^{1}&x_{11}^{3}&x_{1}^{2}&x_{2}^{2}\end{pmatrix}$, $\begin{pmatrix}x_{10}^{2}&x_{11}^{2}&x_{11}^{3}&x_{10}^{3}\\ x_{2}^{2}&x_{4}^{2}&x_{5}^{3}&x_{8}^{3}\end{pmatrix}$,

$\begin{pmatrix}x_{7}^{1}&x_{16}^{1}&x_{16}^{2}&x_{7}^{2}\\ x_{5}^{1}&x_{16}^{3}&x_{12}^{2}&x_{6}^{2}\end{pmatrix}$, $\begin{pmatrix}x_{13}^{2}&x_{16}^{2}&x_{16}^{3}&x_{13}^{3}\\ x_{8}^{2}&x_{6}^{2}&x_{6}^{3}&x_{8}^{3}\end{pmatrix}$.

4.3.3 An $L_{8}$- decomposition of $K_{15}\Box K_{3}$

$\begin{pmatrix}x_{7}^{j}&x_{8}^{j}&x_{9}^{j}&x_{10}^{j}\\ x_{15}^{j}&x_{14}^{j}&x_{1}^{j}&x_{5}^{j}\end{pmatrix}$ for $j=2,3$; $\begin{pmatrix}x_{5}^{j}&x_{13}^{j}&x_{6}^{j}&x_{15}^{j}\\ x_{14}^{j}&x_{4}^{j}&x_{7}^{j}&x_{2}^{j}\end{pmatrix}$ for $j=1,2,3$;

$\begin{pmatrix}x_{3}^{j}&x_{11}^{j}&x_{9}^{j}&x_{12}^{j}\\ x_{8}^{j}&x_{6}^{j}&x_{15}^{j}&x_{10}^{j}\end{pmatrix}$ for $j=1,3$; $\begin{pmatrix}x_{1}^{j}&x_{12}^{j}&x_{2}^{j}&x_{13}^{j}\\ x_{4}^{j}&x_{k}^{j}&x_{7}^{j}&x_{14}^{j}\end{pmatrix}$ for $(j,k)=(1,15),(2,5),(3,15)$;

$\begin{pmatrix}x_{10}^{j}&x_{13}^{j}&x_{11}^{j}&x_{14}^{j}\\ x_{15}^{j}&x_{8}^{j}&x_{k}^{j}&x_{1}^{j}\end{pmatrix}$ for $(j,k)=(1,4),(2,12),(3,4)$;

$\begin{pmatrix}x_{3}^{j}&x_{7}^{j}&x_{4}^{j}&x_{9}^{j}\\ x_{10}^{j}&x_{13}^{j}&x_{k}^{j}&x_{14}^{j}\end{pmatrix}$ for $(j,k)=(1,5),(2,8),(3,8)$;

$\begin{pmatrix}x_{1}^{1}&x_{2}^{1}&x_{14}^{1}&x_{15}^{1}\\ x_{7}^{1}&x_{8}^{1}&x_{12}^{1}&x_{15}^{3}\end{pmatrix}$, $\begin{pmatrix}x_{1}^{2}&x_{2}^{2}&x_{14}^{2}&x_{15}^{2}\\ x_{1}^{3}&x_{8}^{2}&x_{14}^{3}&x_{15}^{3}\end{pmatrix}$, $\begin{pmatrix}x_{1}^{3}&x_{2}^{3}&x_{14}^{3}&x_{15}^{3}\\ x_{7}^{3}&x_{8}^{3}&x_{14}^{1}&x_{13}^{3}\end{pmatrix}$, $\begin{pmatrix}x_{3}^{1}&x_{4}^{1}&x_{12}^{1}&x_{13}^{1}\\ x_{3}^{3}&x_{4}^{3}&x_{5}^{1}&x_{9}^{1}\end{pmatrix}$,

$\begin{pmatrix}x_{3}^{2}&x_{4}^{2}&x_{12}^{2}&x_{13}^{2}\\ x_{3}^{3}&x_{4}^{3}&x_{14}^{2}&x_{13}^{3}\end{pmatrix}$, $\begin{pmatrix}x_{3}^{3}&x_{4}^{3}&x_{12}^{3}&x_{13}^{3}\\ x_{1}^{3}&x_{2}^{3}&x_{5}^{3}&x_{9}^{3}\end{pmatrix}$, $\begin{pmatrix}x_{5}^{1}&x_{6}^{1}&x_{10}^{1}&x_{11}^{1}\\ x_{3}^{1}&x_{6}^{2}&x_{2}^{1}&x_{15}^{1}\end{pmatrix}$, $\begin{pmatrix}x_{5}^{2}&x_{6}^{2}&x_{10}^{2}&x_{11}^{2}\\ x_{3}^{2}&x_{4}^{2}&x_{10}^{3}&x_{11}^{3}\end{pmatrix}$,

$\begin{pmatrix}x_{5}^{3}&x_{6}^{3}&x_{10}^{3}&x_{11}^{3}\\ x_{3}^{3}&x_{4}^{3}&x_{2}^{3}&x_{15}^{3}\end{pmatrix}$, $\begin{pmatrix}x_{7}^{1}&x_{8}^{1}&x_{9}^{1}&x_{10}^{1}\\ x_{7}^{3}&x_{14}^{1}&x_{1}^{1}&x_{5}^{1}\end{pmatrix}$, $\begin{pmatrix}x_{3}^{1}&x_{14}^{1}&x_{4}^{1}&x_{15}^{1}\\ x_{2}^{1}&x_{14}^{2}&x_{10}^{1}&x_{8}^{1}\end{pmatrix}$, $\begin{pmatrix}x_{3}^{2}&x_{14}^{2}&x_{4}^{2}&x_{15}^{2}\\ x_{2}^{2}&x_{7}^{2}&x_{10}^{2}&x_{12}^{2}\end{pmatrix}$,

$\begin{pmatrix}x_{3}^{3}&x_{14}^{3}&x_{4}^{3}&x_{15}^{3}\\ x_{2}^{3}&x_{7}^{3}&x_{10}^{3}&x_{8}^{3}\end{pmatrix}$, $\begin{pmatrix}x_{5}^{1}&x_{8}^{1}&x_{6}^{1}&x_{9}^{1}\\ x_{5}^{2}&x_{1}^{1}&x_{6}^{3}&x_{2}^{1}\end{pmatrix}$, $\begin{pmatrix}x_{5}^{2}&x_{8}^{2}&x_{6}^{2}&x_{9}^{2}\\ x_{4}^{2}&x_{1}^{2}&x_{6}^{3}&x_{9}^{3}\end{pmatrix}$, $\begin{pmatrix}x_{5}^{3}&x_{8}^{3}&x_{6}^{3}&x_{9}^{3}\\ x_{4}^{3}&x_{1}^{3}&x_{3}^{3}&x_{2}^{3}\end{pmatrix}$,

$\begin{pmatrix}x_{7}^{1}&x_{11}^{1}&x_{8}^{1}&x_{12}^{1}\\ x_{14}^{1}&x_{1}^{1}&x_{8}^{3}&x_{6}^{1}\end{pmatrix}$, $\begin{pmatrix}x_{7}^{2}&x_{11}^{2}&x_{8}^{2}&x_{12}^{2}\\ x_{7}^{3}&x_{1}^{2}&x_{8}^{3}&x_{6}^{2}\end{pmatrix}$, $\begin{pmatrix}x_{7}^{3}&x_{11}^{3}&x_{8}^{3}&x_{12}^{3}\\ x_{9}^{3}&x_{1}^{3}&x_{10}^{3}&x_{6}^{3}\end{pmatrix}$, $\begin{pmatrix}x_{1}^{1}&x_{5}^{1}&x_{2}^{1}&x_{6}^{1}\\ x_{10}^{1}&x_{5}^{3}&x_{11}^{1}&x_{14}^{1}\end{pmatrix}$,

$\begin{pmatrix}x_{1}^{2}&x_{5}^{2}&x_{2}^{2}&x_{6}^{2}\\ x_{10}^{2}&x_{5}^{3}&x_{11}^{2}&x_{14}^{2}\end{pmatrix}$, $\begin{pmatrix}x_{1}^{3}&x_{5}^{3}&x_{2}^{3}&x_{6}^{3}\\ x_{10}^{3}&x_{7}^{3}&x_{2}^{2}&x_{14}^{3}\end{pmatrix}$, $\begin{pmatrix}x_{3}^{2}&x_{11}^{2}&x_{9}^{2}&x_{12}^{2}\\ x_{8}^{2}&x_{6}^{2}&x_{15}^{2}&x_{12}^{3}\end{pmatrix}$,

$\begin{pmatrix}x_{11}^{1}&x_{12}^{1}&x_{12}^{3}&x_{11}^{3}\\ x_{11}^{2}&x_{12}^{2}&x_{14}^{3}&x_{2}^{3}\end{pmatrix}$, $\begin{pmatrix}x_{1}^{1}&x_{3}^{1}&x_{3}^{2}&x_{1}^{2}\\ x_{1}^{3}&x_{6}^{1}&x_{6}^{2}&x_{7}^{2}\end{pmatrix}$, $\begin{pmatrix}x_{2}^{1}&x_{4}^{1}&x_{4}^{2}&x_{2}^{2}\\ x_{2}^{3}&x_{6}^{1}&x_{11}^{2}&x_{10}^{2}\end{pmatrix}$,

$\begin{pmatrix}x_{7}^{1}&x_{9}^{1}&x_{9}^{2}&x_{7}^{2}\\ x_{5}^{1}&x_{9}^{3}&x_{2}^{2}&x_{5}^{2}\end{pmatrix}$, $\begin{pmatrix}x_{8}^{1}&x_{10}^{1}&x_{10}^{2}&x_{8}^{2}\\ x_{4}^{1}&x_{10}^{3}&x_{12}^{2}&x_{15}^{2}\end{pmatrix}$, $\begin{pmatrix}x_{13}^{1}&x_{15}^{1}&x_{15}^{2}&x_{13}^{2}\\ x_{13}^{3}&x_{7}^{1}&x_{11}^{2}&x_{9}^{2}\end{pmatrix}$.

4.4 $L_{8}$-decomposition required for Lemma 2.7

4.4.1 An $L_{8}$- decomposition of $K_{16}\Box K_{5}$

$\begin{pmatrix}x_{1}^{j}&x_{2}^{j}&x_{15}^{j}&x_{16}^{j}\\ x_{1}^{5}&x_{2}^{5}&x_{15}^{5}&x_{16}^{5}\end{pmatrix}$, $\begin{pmatrix}x_{3}^{j}&x_{4}^{j}&x_{13}^{j}&x_{14}^{j}\\ x_{3}^{5}&x_{4}^{5}&x_{13}^{5}&x_{14}^{5}\end{pmatrix}$ for $j=1,2,3,4$;

$\begin{pmatrix}x_{9}^{j}&x_{13}^{j}&x_{10}^{j}&x_{14}^{j}\\ x_{3}^{j}&x_{16}^{j}&x_{12}^{j}&x_{7}^{j}\end{pmatrix}$, $\begin{pmatrix}x_{5}^{j}&x_{9}^{j}&x_{6}^{j}&x_{10}^{j}\\ x_{7}^{j}&x_{12}^{j}&x_{8}^{j}&x_{11}^{j}\end{pmatrix}$ for $j=1,2,5$;

$\begin{pmatrix}x_{5}^{j}&x_{6}^{j}&x_{11}^{j}&x_{12}^{j}\\ x_{5}^{5}&x_{6}^{5}&x_{11}^{5}&x_{12}^{5}\end{pmatrix}$, $\begin{pmatrix}x_{7}^{j}&x_{8}^{j}&x_{9}^{j}&x_{10}^{j}\\ x_{7}^{5}&x_{8}^{5}&x_{9}^{5}&x_{10}^{5}\end{pmatrix}$ for $j=1,2,3,4$;

$\begin{pmatrix}x_{12}^{j}&x_{15}^{j}&x_{15}^{k}&x_{12}^{k}\\ x_{2}^{j}&x_{8}^{j}&x_{14}^{k}&x_{9}^{k}\end{pmatrix}$, $\begin{pmatrix}x_{11}^{j}&x_{16}^{j}&x_{16}^{k}&x_{11}^{k}\\ x_{4}^{j}&x_{6}^{j}&x_{13}^{k}&x_{10}^{k}\end{pmatrix}$ for $(j,k)=(1,3),(2,4)$;

$\begin{pmatrix}x_{7}^{j}&x_{11}^{j}&x_{8}^{j}&x_{12}^{j}\\ x_{9}^{j}&x_{15}^{j}&x_{10}^{j}&x_{6}^{j}\end{pmatrix}$, $\begin{pmatrix}x_{1}^{j}&x_{7}^{j}&x_{2}^{j}&x_{8}^{j}\\ x_{4}^{j}&x_{16}^{j}&x_{9}^{j}&x_{13}^{j}\end{pmatrix}$ for $j=1,2,3,4,5$;

$\begin{pmatrix}x_{1}^{j}&x_{5}^{j}&x_{2}^{j}&x_{6}^{j}\\ x_{3}^{j}&x_{8}^{j}&x_{12}^{j}&x_{16}^{j}\end{pmatrix}$ for $j=3,4,5$; $\begin{pmatrix}x_{3}^{j}&x_{7}^{j}&x_{4}^{j}&x_{8}^{j}\\ x_{13}^{j}&x_{7}^{k}&x_{14}^{j}&x_{16}^{j}\end{pmatrix}$ for $(j,k)=(3,1),(4,2)$;

$\begin{pmatrix}x_{1}^{j}&x_{13}^{j}&x_{2}^{j}&x_{14}^{j}\\ x_{k_{1}}^{k_{2}}&x_{6}^{j}&x_{10}^{j}&x_{5}^{j}\end{pmatrix}$ for $j=k_{2}=3,4,5,k_{1}=11\&(j,k_{1},k_{2})=(1,1,3),(2,1,4)$;

$\begin{pmatrix}x_{3}^{j}&x_{15}^{j}&x_{4}^{j}&x_{16}^{j}\\ x_{k_{1}}^{k_{2}}&x_{1}^{j}&x_{10}^{j}&x_{2}^{j}\end{pmatrix}$ for $j=k_{2}=1,2,5,k_{1}=11\&(j,k_{1},k_{2})=(3,3,1),(4,3,2)$;

$\begin{pmatrix}x_{3}^{j}&x_{5}^{j}&x_{4}^{j}&x_{6}^{j}\\ x_{2}^{j}&x_{11}^{j}&x_{k_{1}}^{k_{2}}&x_{14}^{j}\end{pmatrix}$ for $j=k_{2}=3,4,5,k_{1}=9\&(j,k_{1},k_{2})=(1,4,3),(2,4,4)$;

$\begin{pmatrix}x_{9}^{j}&x_{15}^{j}&x_{10}^{j}&x_{16}^{j}\\ x_{1}^{j}&x_{k}^{j}&x_{3}^{j}&x_{14}^{j}\end{pmatrix}$ for $(j,k)=(1,5),(2,5),(3,7),(4,7)$;

$\begin{pmatrix}x_{11}^{j}&x_{13}^{j}&x_{12}^{j}&x_{14}^{j}\\ x_{k}^{j}&x_{15}^{j}&x_{4}^{j}&x_{8}^{j}\end{pmatrix}$ for $(j,k)=(1,1),(2,1),(3,3),(4,3)$;

$\begin{pmatrix}x_{3}^{j}&x_{7}^{j}&x_{4}^{j}&x_{8}^{j}\\ x_{13}^{j}&x_{15}^{j}&x_{14}^{j}&x_{8}^{k}\end{pmatrix}$ for $(j,k)=(1,3),(2,4)$;

$\begin{pmatrix}x_{i}^{1}&x_{i}^{2}&x_{i}^{3}&x_{i}^{4}\\ x_{k}^{1}&x_{k}^{2}&x_{k}^{3}&x_{k}^{4}\end{pmatrix}$ for $(i,k)=(1,12),(2,4),(3,12),(6,7)(7,13),(9,11),(10,1),\\ (11,2),(12,16),(13,5),(15,6),(16,5)$;

$\begin{pmatrix}x_{i}^{1}&x_{i}^{2}&x_{i}^{3}&x_{i}^{4}\\ x_{k_{1}}^{1}&x_{k_{1}}^{2}&x_{k_{2}}^{3}&x_{k_{2}}^{4}\end{pmatrix}$ for $(i,k_{1},k_{2})=(4,9,11),(5,8,15),(8,16,15),(14,15,7)$;

$\begin{pmatrix}x_{1}^{5}&x_{2}^{5}&x_{15}^{5}&x_{16}^{5}\\ x_{12}^{5}&x_{4}^{5}&x_{6}^{5}&x_{5}^{5}\end{pmatrix}$, $\begin{pmatrix}x_{3}^{5}&x_{4}^{5}&x_{13}^{5}&x_{14}^{5}\\ x_{12}^{5}&x_{11}^{5}&x_{5}^{5}&x_{15}^{5}\end{pmatrix}$, $\begin{pmatrix}x_{5}^{5}&x_{6}^{5}&x_{11}^{5}&x_{12}^{5}\\ x_{15}^{5}&x_{7}^{5}&x_{2}^{5}&x_{16}^{5}\end{pmatrix}$, $\begin{pmatrix}x_{7}^{5}&x_{8}^{5}&x_{9}^{5}&x_{10}^{5}\\ x_{13}^{5}&x_{15}^{5}&x_{11}^{5}&x_{1}^{5}\end{pmatrix}$,

$\begin{pmatrix}x_{5}^{3}&x_{9}^{3}&x_{6}^{3}&x_{10}^{3}\\ x_{7}^{3}&x_{9}^{1}&x_{8}^{3}&x_{10}^{1}\end{pmatrix}$, $\begin{pmatrix}x_{5}^{4}&x_{9}^{4}&x_{6}^{4}&x_{10}^{4}\\ x_{7}^{4}&x_{9}^{2}&x_{8}^{4}&x_{10}^{2}\end{pmatrix}$, $\begin{pmatrix}x_{9}^{5}&x_{15}^{5}&x_{10}^{5}&x_{16}^{5}\\ x_{1}^{5}&x_{12}^{5}&x_{3}^{5}&x_{14}^{5}\end{pmatrix}$, $\begin{pmatrix}x_{11}^{5}&x_{13}^{5}&x_{12}^{5}&x_{14}^{5}\\ x_{16}^{5}&x_{15}^{5}&x_{4}^{5}&x_{8}^{5}\end{pmatrix}$,

$\begin{pmatrix}x_{3}^{5}&x_{7}^{5}&x_{4}^{5}&x_{8}^{5}\\ x_{13}^{5}&x_{15}^{5}&x_{14}^{5}&x_{16}^{5}\end{pmatrix}$, $\begin{pmatrix}x_{1}^{1}&x_{5}^{1}&x_{2}^{1}&x_{6}^{1}\\ x_{3}^{1}&x_{5}^{3}&x_{2}^{3}&x_{6}^{3}\end{pmatrix}$, $\begin{pmatrix}x_{1}^{2}&x_{5}^{2}&x_{2}^{2}&x_{6}^{2}\\ x_{3}^{2}&x_{5}^{4}&x_{2}^{4}&x_{6}^{4}\end{pmatrix}$, $\begin{pmatrix}x_{9}^{3}&x_{13}^{3}&x_{10}^{3}&x_{14}^{3}\\ x_{3}^{3}&x_{13}^{1}&x_{12}^{3}&x_{14}^{1}\end{pmatrix}$,

$\begin{pmatrix}x_{9}^{4}&x_{13}^{4}&x_{10}^{4}&x_{14}^{4}\\ x_{3}^{4}&x_{13}^{2}&x_{12}^{4}&x_{14}^{2}\end{pmatrix}$.

4.4.2 An $L_{8}$- decomposition of $K_{13}\Box K_{5}$

$\begin{pmatrix}x_{5}^{j}&x_{9}^{j}&x_{6}^{j}&x_{10}^{j}\\ x_{3}^{j}&x_{9}^{3}&x_{4}^{j}&x_{10}^{3}\end{pmatrix}$, $\begin{pmatrix}x_{3}^{j}&x_{12}^{j}&x_{4}^{j}&x_{13}^{j}\\ x_{11}^{j}&x_{2}^{j}&x_{9}^{j}&x_{7}^{j}\end{pmatrix}$ for $j=2,4$;

$\begin{pmatrix}x_{7}^{j}&x_{8}^{j}&x_{10}^{j}&x_{11}^{j}\\ x_{9}^{j}&x_{2}^{j}&x_{1}^{j}&x_{13}^{j}\end{pmatrix}$ for $j=1,5$; $\begin{pmatrix}x_{1}^{j}&x_{13}^{j}&x_{8}^{j}&x_{12}^{j}\\ x_{7}^{j}&x_{13}^{3}&x_{9}^{j}&x_{12}^{3}\end{pmatrix}$ for $j=2,5$;

$\begin{pmatrix}x_{9}^{j}&x_{12}^{j}&x_{10}^{j}&x_{13}^{j}\\ x_{8}^{j}&x_{6}^{j}&x_{4}^{j}&x_{5}^{j}\end{pmatrix}$ for $j=3,4$; $\begin{pmatrix}x_{i}^{1}&x_{i}^{2}&x_{i}^{4}&x_{i}^{3}\\ x_{k}^{1}&x_{k}^{2}&x_{k}^{4}&x_{k}^{3}\end{pmatrix}$ for $(i,k)=(5,7),(8,11)$;

$\begin{pmatrix}x_{1}^{j}&x_{5}^{j}&x_{2}^{j}&x_{6}^{j}\\ x_{k_{1}}^{j}&x_{5}^{k_{2}}&x_{2}^{k_{2}}&x_{6}^{k_{2}}\end{pmatrix}$ for $(j,k_{1},k_{2})=(1,9,4),(2,8,5),(3,9,2)$;

$\begin{pmatrix}x_{i}^{1}&x_{i}^{2}&x_{i}^{5}&x_{i}^{4}\\ x_{k_{1}}^{1}&x_{k_{2}}^{2}&x_{k_{1}}^{5}&x_{k_{2}}^{4}\end{pmatrix}$ for $(i,k_{1},k_{2})=(9,10,11),(10,2,1),(13,6,6)$;

$\begin{pmatrix}x_{1}^{4}&x_{5}^{4}&x_{2}^{4}&x_{6}^{4}\\ x_{1}^{5}&x_{5}^{5}&x_{11}^{4}&x_{6}^{5}\end{pmatrix}$, $\begin{pmatrix}x_{1}^{5}&x_{5}^{5}&x_{2}^{5}&x_{6}^{5}\\ x_{1}^{3}&x_{5}^{3}&x_{2}^{3}&x_{6}^{3}\end{pmatrix}$, $\begin{pmatrix}x_{3}^{1}&x_{7}^{1}&x_{4}^{1}&x_{8}^{1}\\ x_{3}^{3}&x_{7}^{5}&x_{4}^{3}&x_{8}^{4}\end{pmatrix}$, $\begin{pmatrix}x_{3}^{2}&x_{7}^{2}&x_{4}^{2}&x_{8}^{2}\\ x_{1}^{2}&x_{7}^{4}&x_{4}^{3}&x_{8}^{5}\end{pmatrix}$,

$\begin{pmatrix}x_{3}^{3}&x_{7}^{3}&x_{4}^{3}&x_{8}^{3}\\ x_{9}^{3}&x_{1}^{3}&x_{2}^{3}&x_{8}^{2}\end{pmatrix}$, $\begin{pmatrix}x_{3}^{4}&x_{7}^{4}&x_{4}^{4}&x_{8}^{4}\\ x_{3}^{2}&x_{7}^{3}&x_{4}^{3}&x_{8}^{5}\end{pmatrix}$, $\begin{pmatrix}x_{3}^{5}&x_{7}^{5}&x_{4}^{5}&x_{8}^{5}\\ x_{3}^{3}&x_{5}^{5}&x_{11}^{5}&x_{8}^{3}\end{pmatrix}$, $\begin{pmatrix}x_{5}^{1}&x_{9}^{1}&x_{6}^{1}&x_{10}^{1}\\ x_{5}^{5}&x_{9}^{3}&x_{12}^{1}&x_{10}^{3}\end{pmatrix}$,

$\begin{pmatrix}x_{5}^{3}&x_{9}^{3}&x_{6}^{3}&x_{10}^{3}\\ x_{3}^{3}&x_{7}^{3}&x_{4}^{3}&x_{1}^{3}\end{pmatrix}$, $\begin{pmatrix}x_{5}^{5}&x_{9}^{5}&x_{6}^{5}&x_{10}^{5}\\ x_{13}^{5}&x_{9}^{3}&x_{6}^{1}&x_{10}^{3}\end{pmatrix}$, $\begin{pmatrix}x_{1}^{1}&x_{4}^{1}&x_{5}^{1}&x_{11}^{1}\\ x_{2}^{1}&x_{6}^{1}&x_{8}^{1}&x_{11}^{3}\end{pmatrix}$, $\begin{pmatrix}x_{1}^{2}&x_{4}^{2}&x_{5}^{2}&x_{11}^{2}\\ x_{2}^{2}&x_{4}^{4}&x_{8}^{2}&x_{11}^{5}\end{pmatrix}$,

$\begin{pmatrix}x_{1}^{3}&x_{4}^{3}&x_{5}^{3}&x_{11}^{3}\\ x_{1}^{4}&x_{13}^{3}&x_{8}^{3}&x_{6}^{3}\end{pmatrix}$, $\begin{pmatrix}x_{1}^{4}&x_{4}^{4}&x_{5}^{4}&x_{11}^{4}\\ x_{2}^{4}&x_{3}^{4}&x_{8}^{4}&x_{11}^{1}\end{pmatrix}$, $\begin{pmatrix}x_{1}^{5}&x_{4}^{5}&x_{5}^{5}&x_{11}^{5}\\ x_{2}^{5}&x_{6}^{5}&x_{8}^{5}&x_{3}^{5}\end{pmatrix}$, $\begin{pmatrix}x_{2}^{1}&x_{3}^{1}&x_{6}^{1}&x_{7}^{1}\\ x_{11}^{1}&x_{4}^{1}&x_{5}^{1}&x_{13}^{1}\end{pmatrix}$,

$\begin{pmatrix}x_{2}^{2}&x_{3}^{2}&x_{6}^{2}&x_{7}^{2}\\ x_{11}^{2}&x_{4}^{2}&x_{5}^{2}&x_{10}^{2}\end{pmatrix}$, $\begin{pmatrix}x_{2}^{3}&x_{3}^{3}&x_{6}^{3}&x_{7}^{3}\\ x_{11}^{3}&x_{3}^{2}&x_{5}^{3}&x_{10}^{3}\end{pmatrix}$, $\begin{pmatrix}x_{2}^{4}&x_{3}^{4}&x_{6}^{4}&x_{7}^{4}\\ x_{9}^{4}&x_{1}^{4}&x_{5}^{4}&x_{10}^{4}\end{pmatrix}$, $\begin{pmatrix}x_{2}^{5}&x_{3}^{5}&x_{6}^{5}&x_{7}^{5}\\ x_{11}^{5}&x_{4}^{5}&x_{5}^{5}&x_{7}^{3}\end{pmatrix}$,

$\begin{pmatrix}x_{7}^{2}&x_{8}^{2}&x_{10}^{2}&x_{11}^{2}\\ x_{9}^{2}&x_{2}^{2}&x_{3}^{2}&x_{4}^{2}\end{pmatrix}$, $\begin{pmatrix}x_{7}^{3}&x_{8}^{3}&x_{10}^{3}&x_{11}^{3}\\ x_{7}^{2}&x_{1}^{3}&x_{3}^{3}&x_{11}^{4}\end{pmatrix}$, $\begin{pmatrix}x_{7}^{4}&x_{8}^{4}&x_{10}^{4}&x_{11}^{4}\\ x_{9}^{4}&x_{1}^{4}&x_{3}^{4}&x_{4}^{4}\end{pmatrix}$, $\begin{pmatrix}x_{1}^{1}&x_{13}^{1}&x_{8}^{1}&x_{12}^{1}\\ x_{7}^{1}&x_{13}^{3}&x_{9}^{1}&x_{5}^{1}\end{pmatrix}$,

$\begin{pmatrix}x_{1}^{3}&x_{13}^{3}&x_{8}^{3}&x_{12}^{3}\\ x_{3}^{3}&x_{6}^{3}&x_{2}^{3}&x_{7}^{3}\end{pmatrix}$, $\begin{pmatrix}x_{1}^{4}&x_{13}^{4}&x_{8}^{4}&x_{12}^{4}\\ x_{7}^{4}&x_{13}^{3}&x_{2}^{4}&x_{12}^{3}\end{pmatrix}$, $\begin{pmatrix}x_{3}^{1}&x_{12}^{1}&x_{4}^{1}&x_{13}^{1}\\ x_{11}^{1}&x_{2}^{1}&x_{4}^{5}&x_{13}^{5}\end{pmatrix}$, $\begin{pmatrix}x_{3}^{3}&x_{4}^{3}&x_{12}^{3}&x_{13}^{3}\\ x_{11}^{3}&x_{9}^{3}&x_{5}^{3}&x_{7}^{3}\end{pmatrix}$,

$\begin{pmatrix}x_{3}^{5}&x_{12}^{5}&x_{4}^{5}&x_{13}^{5}\\ x_{1}^{5}&x_{2}^{5}&x_{4}^{3}&x_{7}^{5}\end{pmatrix}$, $\begin{pmatrix}x_{9}^{1}&x_{12}^{1}&x_{10}^{1}&x_{13}^{1}\\ x_{11}^{1}&x_{12}^{3}&x_{4}^{1}&x_{5}^{1}\end{pmatrix}$, $\begin{pmatrix}x_{9}^{2}&x_{12}^{2}&x_{10}^{2}&x_{13}^{2}\\ x_{2}^{2}&x_{6}^{2}&x_{4}^{2}&x_{5}^{2}\end{pmatrix}$, $\begin{pmatrix}x_{9}^{5}&x_{12}^{5}&x_{10}^{5}&x_{13}^{5}\\ x_{11}^{5}&x_{6}^{5}&x_{4}^{5}&x_{2}^{5}\end{pmatrix}$,

$\begin{pmatrix}x_{1}^{1}&x_{1}^{3}&x_{1}^{2}&x_{1}^{4}\\ x_{3}^{1}&x_{2}^{3}&x_{1}^{5}&x_{9}^{4}\end{pmatrix}$, $\begin{pmatrix}x_{2}^{1}&x_{2}^{2}&x_{2}^{4}&x_{2}^{3}\\ x_{4}^{1}&x_{13}^{2}&x_{13}^{4}&x_{10}^{3}\end{pmatrix}$, $\begin{pmatrix}x_{3}^{1}&x_{3}^{2}&x_{3}^{5}&x_{3}^{4}\\ x_{9}^{1}&x_{9}^{2}&x_{9}^{5}&x_{3}^{3}\end{pmatrix}$, $\begin{pmatrix}x_{4}^{1}&x_{4}^{2}&x_{4}^{5}&x_{4}^{4}\\ x_{11}^{1}&x_{2}^{2}&x_{2}^{5}&x_{2}^{4}\end{pmatrix}$,

$\begin{pmatrix}x_{6}^{1}&x_{6}^{2}&x_{6}^{4}&x_{6}^{3}\\ x_{11}^{1}&x_{8}^{2}&x_{8}^{4}&x_{8}^{3}\end{pmatrix}$, $\begin{pmatrix}x_{7}^{1}&x_{7}^{2}&x_{7}^{5}&x_{7}^{4}\\ x_{7}^{3}&x_{12}^{2}&x_{12}^{5}&x_{12}^{4}\end{pmatrix}$, $\begin{pmatrix}x_{11}^{2}&x_{11}^{3}&x_{11}^{5}&x_{11}^{4}\\ x_{6}^{2}&x_{13}^{3}&x_{6}^{5}&x_{6}^{4}\end{pmatrix}$, $\begin{pmatrix}x_{12}^{1}&x_{12}^{2}&x_{12}^{5}&x_{12}^{4}\\ x_{7}^{1}&x_{11}^{2}&x_{5}^{5}&x_{11}^{4}\end{pmatrix}$,

$\begin{pmatrix}x_{2}^{1}&x_{2}^{5}&x_{9}^{5}&x_{9}^{1}\\ x_{13}^{1}&x_{2}^{4}&x_{4}^{5}&x_{4}^{1}\end{pmatrix}$, $\begin{pmatrix}x_{9}^{2}&x_{9}^{4}&x_{10}^{4}&x_{10}^{2}\\ x_{1}^{2}&x_{3}^{4}&x_{2}^{4}&x_{2}^{2}\end{pmatrix}$, $\begin{pmatrix}x_{11}^{1}&x_{11}^{5}&x_{12}^{5}&x_{12}^{1}\\ x_{11}^{2}&x_{8}^{5}&x_{13}^{5}&x_{13}^{1}\end{pmatrix}$, $\begin{pmatrix}x_{12}^{2}&x_{12}^{4}&x_{13}^{4}&x_{13}^{2}\\ x_{5}^{2}&x_{5}^{4}&x_{11}^{4}&x_{11}^{2}\end{pmatrix}$,

$\begin{pmatrix}x_{2}^{3}&x_{9}^{3}&x_{11}^{3}&x_{12}^{3}\\ x_{13}^{3}&x_{10}^{3}&x_{4}^{3}&x_{3}^{3}\end{pmatrix}$, $\begin{pmatrix}x_{3}^{1}&x_{3}^{5}&x_{10}^{5}&x_{10}^{1}\\ x_{5}^{1}&x_{5}^{5}&x_{7}^{5}&x_{7}^{1}\end{pmatrix}$, $\begin{pmatrix}x_{1}^{1}&x_{1}^{5}&x_{8}^{5}&x_{8}^{1}\\ x_{1}^{2}&x_{9}^{5}&x_{6}^{5}&x_{6}^{1}\end{pmatrix}$.

4.5 $L_{8}$-decomposition required for Lemma 2.8

4.5.1 An $L_{8}$- decomposition of $K_{11}\Box K_{7}$

$\begin{pmatrix}x_{1}^{j}&x_{5}^{j}&x_{2}^{j}&x_{6}^{j}\\ x_{9}^{j}&x_{7}^{j}&x_{4}^{j}&x_{8}^{j}\end{pmatrix}$ for $j=2,5$; $\begin{pmatrix}x_{1}^{j}&x_{4}^{j}&x_{5}^{j}&x_{11}^{j}\\ x_{3}^{j}&x_{9}^{j}&x_{8}^{j}&x_{11}^{k}\end{pmatrix}$ for $(j,k)=(2,3),(3,4)$;

$\begin{pmatrix}x_{5}^{j}&x_{9}^{j}&x_{6}^{j}&x_{10}^{j}\\ x_{3}^{j}&x_{11}^{j}&x_{4}^{j}&x_{1}^{j}\end{pmatrix}$ for $j=5,6$; $\begin{pmatrix}x_{7}^{j}&x_{8}^{j}&x_{10}^{j}&x_{11}^{j}\\ x_{9}^{j}&x_{1}^{j}&x_{2}^{j}&x_{4}^{j}\end{pmatrix}$ for $j=1,2,5$;

$\begin{pmatrix}x_{2}^{j}&x_{3}^{j}&x_{6}^{j}&x_{7}^{j}\\ x_{1}^{j}&x_{11}^{j}&x_{5}^{j}&x_{7}^{k}\end{pmatrix}$ for $(j,k)=(5,2),(6,2)$;

$\begin{pmatrix}x_{7}^{j}&x_{8}^{j}&x_{10}^{j}&x_{11}^{j}\\ x_{9}^{j}&x_{1}^{j}&x_{10}^{k}&x_{4}^{j}\end{pmatrix}$ for $(j,k)=(3,6),(6,7)$;

$\begin{pmatrix}x_{i}^{1}&x_{i}^{3}&x_{i}^{5}&x_{i}^{7}\\ x_{i}^{4}&x_{i}^{2}&x_{i}^{6}&x_{k}^{7}\end{pmatrix}$ for $(i,k)=(5,9),(6,8),(7,1),(8,9)$;

$\begin{pmatrix}x_{i}^{1}&x_{i}^{3}&x_{i}^{5}&x_{i}^{7}\\ x_{k_{1}}^{1}&x_{i}^{2}&x_{i}^{6}&x_{k_{2}}^{7}\end{pmatrix}$ for $(i,k_{1},k_{2})=(1,9,9),(4,3,3),(9,8,2)$;

$\begin{pmatrix}x_{i}^{1}&x_{i}^{2}&x_{i}^{4}&x_{i}^{6}\\ x_{i}^{5}&x_{k}^{2}&x_{i}^{3}&x_{i}^{7}\end{pmatrix}$ for $(i,k)=(4,3),(8,9),(9,2)$;

$\begin{pmatrix}x_{i}^{2}&x_{i}^{5}&x_{i}^{4}&x_{i}^{7}\\ x_{i}^{6}&x_{k}^{5}&x_{k}^{4}&x_{i}^{3}\end{pmatrix}$ for $(i,k)=(3,10),(8,9)$;

$\begin{pmatrix}x_{1}^{1}&x_{5}^{1}&x_{2}^{1}&x_{6}^{1}\\ x_{10}^{1}&x_{3}^{1}&x_{9}^{1}&x_{8}^{1}\end{pmatrix}$, $\begin{pmatrix}x_{1}^{3}&x_{5}^{3}&x_{2}^{3}&x_{6}^{3}\\ x_{9}^{3}&x_{7}^{3}&x_{10}^{3}&x_{11}^{3}\end{pmatrix}$, $\begin{pmatrix}x_{1}^{4}&x_{5}^{4}&x_{2}^{4}&x_{6}^{4}\\ x_{1}^{1}&x_{10}^{4}&x_{2}^{1}&x_{8}^{4}\end{pmatrix}$, $\begin{pmatrix}x_{1}^{6}&x_{5}^{6}&x_{2}^{6}&x_{6}^{6}\\ x_{1}^{3}&x_{5}^{3}&x_{2}^{2}&x_{6}^{3}\end{pmatrix}$,

$\begin{pmatrix}x_{1}^{7}&x_{5}^{7}&x_{2}^{7}&x_{6}^{7}\\ x_{1}^{6}&x_{5}^{6}&x_{2}^{6}&x_{11}^{7}\end{pmatrix}$, $\begin{pmatrix}x_{3}^{1}&x_{7}^{1}&x_{4}^{1}&x_{8}^{1}\\ x_{11}^{1}&x_{1}^{1}&x_{10}^{1}&x_{2}^{1}\end{pmatrix}$, $\begin{pmatrix}x_{3}^{2}&x_{7}^{2}&x_{4}^{2}&x_{8}^{2}\\ x_{9}^{2}&x_{1}^{2}&x_{10}^{2}&x_{2}^{2}\end{pmatrix}$, $\begin{pmatrix}x_{3}^{3}&x_{7}^{3}&x_{4}^{3}&x_{8}^{3}\\ x_{5}^{3}&x_{1}^{3}&x_{10}^{3}&x_{2}^{3}\end{pmatrix}$,

$\begin{pmatrix}x_{3}^{4}&x_{7}^{4}&x_{4}^{4}&x_{8}^{4}\\ x_{3}^{3}&x_{1}^{4}&x_{9}^{4}&x_{2}^{4}\end{pmatrix}$, $\begin{pmatrix}x_{3}^{5}&x_{7}^{5}&x_{4}^{5}&x_{8}^{5}\\ x_{9}^{5}&x_{1}^{5}&x_{10}^{5}&x_{5}^{5}\end{pmatrix}$, $\begin{pmatrix}x_{3}^{6}&x_{7}^{6}&x_{4}^{6}&x_{8}^{6}\\ x_{9}^{6}&x_{7}^{3}&x_{10}^{6}&x_{5}^{6}\end{pmatrix}$, $\begin{pmatrix}x_{3}^{7}&x_{7}^{7}&x_{4}^{7}&x_{8}^{7}\\ x_{5}^{7}&x_{7}^{6}&x_{6}^{7}&x_{2}^{7}\end{pmatrix}$,

$\begin{pmatrix}x_{5}^{1}&x_{9}^{1}&x_{6}^{1}&x_{10}^{1}\\ x_{8}^{1}&x_{11}^{1}&x_{4}^{1}&x_{10}^{4}\end{pmatrix}$, $\begin{pmatrix}x_{5}^{2}&x_{9}^{2}&x_{6}^{2}&x_{10}^{2}\\ x_{3}^{2}&x_{11}^{2}&x_{4}^{2}&x_{10}^{3}\end{pmatrix}$, $\begin{pmatrix}x_{5}^{3}&x_{9}^{3}&x_{6}^{3}&x_{10}^{3}\\ x_{5}^{7}&x_{9}^{6}&x_{4}^{3}&x_{1}^{3}\end{pmatrix}$, $\begin{pmatrix}x_{7}^{7}&x_{8}^{7}&x_{10}^{7}&x_{11}^{7}\\ x_{9}^{7}&x_{1}^{7}&x_{2}^{7}&x_{3}^{7}\end{pmatrix}$,

$\begin{pmatrix}x_{1}^{1}&x_{4}^{1}&x_{5}^{1}&x_{11}^{1}\\ x_{3}^{1}&x_{9}^{1}&x_{7}^{1}&x_{8}^{1}\end{pmatrix}$, $\begin{pmatrix}x_{1}^{4}&x_{4}^{4}&x_{5}^{4}&x_{11}^{4}\\ x_{3}^{4}&x_{10}^{4}&x_{8}^{4}&x_{9}^{4}\end{pmatrix}$, $\begin{pmatrix}x_{1}^{5}&x_{4}^{5}&x_{5}^{5}&x_{11}^{5}\\ x_{3}^{5}&x_{9}^{5}&x_{5}^{2}&x_{6}^{5}\end{pmatrix}$, $\begin{pmatrix}x_{1}^{6}&x_{4}^{6}&x_{5}^{6}&x_{11}^{6}\\ x_{3}^{1}&x_{9}^{6}&x_{5}^{2}&x_{11}^{2}\end{pmatrix}$,

$\begin{pmatrix}x_{1}^{7}&x_{4}^{7}&x_{5}^{7}&x_{11}^{7}\\ x_{3}^{7}&x_{2}^{7}&x_{8}^{7}&x_{11}^{6}\end{pmatrix}$, $\begin{pmatrix}x_{2}^{1}&x_{3}^{1}&x_{6}^{1}&x_{7}^{1}\\ x_{11}^{1}&x_{9}^{1}&x_{5}^{1}&x_{10}^{1}\end{pmatrix}$, $\begin{pmatrix}x_{2}^{2}&x_{3}^{2}&x_{6}^{2}&x_{7}^{2}\\ x_{1}^{2}&x_{3}^{3}&x_{5}^{2}&x_{10}^{2}\end{pmatrix}$, $\begin{pmatrix}x_{2}^{3}&x_{3}^{3}&x_{6}^{3}&x_{7}^{3}\\ x_{1}^{3}&x_{9}^{3}&x_{5}^{3}&x_{10}^{3}\end{pmatrix}$,

$\begin{pmatrix}x_{2}^{4}&x_{3}^{4}&x_{6}^{4}&x_{7}^{4}\\ x_{1}^{4}&x_{11}^{4}&x_{9}^{4}&x_{10}^{4}\end{pmatrix}$, $\begin{pmatrix}x_{2}^{7}&x_{3}^{7}&x_{6}^{7}&x_{7}^{7}\\ x_{2}^{3}&x_{9}^{7}&x_{5}^{7}&x_{10}^{7}\end{pmatrix}$, $\begin{pmatrix}x_{2}^{1}&x_{2}^{3}&x_{2}^{5}&x_{2}^{7}\\ x_{4}^{1}&x_{2}^{4}&x_{8}^{5}&x_{11}^{7}\end{pmatrix}$, $\begin{pmatrix}x_{3}^{1}&x_{3}^{3}&x_{3}^{5}&x_{3}^{7}\\ x_{10}^{1}&x_{4}^{3}&x_{3}^{6}&x_{10}^{7}\end{pmatrix}$,

$\begin{pmatrix}x_{10}^{1}&x_{10}^{3}&x_{10}^{5}&x_{10}^{7}\\ x_{9}^{1}&x_{3}^{3}&x_{7}^{5}&x_{9}^{7}\end{pmatrix}$, $\begin{pmatrix}x_{11}^{1}&x_{11}^{3}&x_{11}^{5}&x_{11}^{7}\\ x_{11}^{4}&x_{9}^{3}&x_{11}^{6}&x_{8}^{7}\end{pmatrix}$, $\begin{pmatrix}x_{1}^{1}&x_{1}^{2}&x_{1}^{4}&x_{1}^{6}\\ x_{2}^{1}&x_{10}^{2}&x_{1}^{3}&x_{9}^{6}\end{pmatrix}$, $\begin{pmatrix}x_{2}^{1}&x_{2}^{2}&x_{2}^{4}&x_{2}^{6}\\ x_{2}^{5}&x_{11}^{2}&x_{9}^{4}&x_{8}^{6}\end{pmatrix}$,

$\begin{pmatrix}x_{3}^{1}&x_{3}^{2}&x_{3}^{4}&x_{3}^{6}\\ x_{3}^{5}&x_{10}^{2}&x_{9}^{4}&x_{3}^{7}\end{pmatrix}$, $\begin{pmatrix}x_{5}^{1}&x_{5}^{2}&x_{5}^{4}&x_{5}^{6}\\ x_{5}^{5}&x_{5}^{7}&x_{5}^{3}&x_{7}^{6}\end{pmatrix}$, $\begin{pmatrix}x_{6}^{1}&x_{6}^{2}&x_{6}^{4}&x_{6}^{6}\\ x_{11}^{1}&x_{11}^{2}&x_{6}^{3}&x_{8}^{6}\end{pmatrix}$, $\begin{pmatrix}x_{7}^{1}&x_{7}^{2}&x_{7}^{4}&x_{7}^{6}\\ x_{7}^{5}&x_{7}^{7}&x_{7}^{3}&x_{1}^{6}\end{pmatrix}$,

$\begin{pmatrix}x_{10}^{1}&x_{10}^{2}&x_{10}^{4}&x_{10}^{6}\\ x_{10}^{5}&x_{9}^{2}&x_{10}^{3}&x_{7}^{6}\end{pmatrix}$, $\begin{pmatrix}x_{11}^{1}&x_{11}^{2}&x_{11}^{4}&x_{11}^{6}\\ x_{11}^{5}&x_{8}^{2}&x_{10}^{4}&x_{6}^{6}\end{pmatrix}$, $\begin{pmatrix}x_{1}^{2}&x_{1}^{5}&x_{1}^{4}&x_{1}^{7}\\ x_{1}^{6}&x_{1}^{1}&x_{9}^{4}&x_{1}^{3}\end{pmatrix}$, $\begin{pmatrix}x_{2}^{2}&x_{2}^{5}&x_{2}^{4}&x_{2}^{7}\\ x_{2}^{3}&x_{11}^{5}&x_{11}^{4}&x_{1}^{7}\end{pmatrix}$,

$\begin{pmatrix}x_{4}^{2}&x_{4}^{5}&x_{4}^{4}&x_{4}^{7}\\ x_{4}^{6}&x_{3}^{5}&x_{4}^{1}&x_{4}^{3}\end{pmatrix}$, $\begin{pmatrix}x_{2}^{6}&x_{4}^{6}&x_{3}^{6}&x_{10}^{6}\\ x_{2}^{5}&x_{4}^{3}&x_{3}^{3}&x_{10}^{5}\end{pmatrix}$, $\begin{pmatrix}x_{6}^{2}&x_{6}^{5}&x_{6}^{4}&x_{6}^{7}\\ x_{6}^{6}&x_{6}^{1}&x_{11}^{4}&x_{6}^{3}\end{pmatrix}$, $\begin{pmatrix}x_{9}^{2}&x_{9}^{5}&x_{9}^{4}&x_{9}^{7}\\ x_{9}^{6}&x_{2}^{5}&x_{9}^{1}&x_{11}^{7}\end{pmatrix}$,

$\begin{pmatrix}x_{10}^{2}&x_{10}^{5}&x_{10}^{4}&x_{10}^{7}\\ x_{10}^{6}&x_{9}^{5}&x_{1}^{4}&x_{10}^{3}\end{pmatrix}$, $\begin{pmatrix}x_{11}^{2}&x_{11}^{5}&x_{11}^{4}&x_{11}^{7}\\ x_{3}^{2}&x_{8}^{5}&x_{8}^{4}&x_{11}^{3}\end{pmatrix}$, $\begin{pmatrix}x_{2}^{3}&x_{9}^{3}&x_{8}^{3}&x_{11}^{3}\\ x_{4}^{3}&x_{10}^{3}&x_{6}^{3}&x_{3}^{3}\end{pmatrix}$, $\begin{pmatrix}x_{2}^{6}&x_{9}^{6}&x_{8}^{6}&x_{11}^{6}\\ x_{2}^{3}&x_{10}^{6}&x_{8}^{3}&x_{11}^{3}\end{pmatrix}$

$\begin{pmatrix}x_{3}^{4}&x_{4}^{4}&x_{6}^{4}&x_{5}^{4}\\ x_{3}^{1}&x_{4}^{1}&x_{10}^{4}&x_{9}^{4}\end{pmatrix}$, $\begin{pmatrix}x_{7}^{4}&x_{8}^{4}&x_{10}^{4}&x_{9}^{4}\\ x_{11}^{4}&x_{1}^{4}&x_{2}^{4}&x_{4}^{4}\end{pmatrix}$, $\begin{pmatrix}x_{4}^{7}&x_{9}^{7}&x_{6}^{7}&x_{10}^{7}\\ x_{11}^{7}&x_{9}^{3}&x_{6}^{6}&x_{1}^{7}\end{pmatrix}$, $\begin{pmatrix}x_{5}^{4}&x_{7}^{4}&x_{7}^{7}&x_{5}^{7}\\ x_{5}^{5}&x_{7}^{5}&x_{7}^{3}&x_{10}^{7}\end{pmatrix}$.

4.6 $L_{8}$-decomposition required for Lemma 2.9

4.6.1 An $L_{8}$- decomposition of $K_{9}\Box K_{9}$

$\begin{pmatrix}x_{1}^{j}&x_{2}^{j}&x_{3}^{j}&x_{4}^{j}\\ x_{7}^{j}&x_{5}^{j}&x_{6}^{j}&x_{8}^{j}\end{pmatrix}$ for $j=4,6,7,8$; $\begin{pmatrix}x_{5}^{j}&x_{6}^{j}&x_{7}^{j}&x_{8}^{j}\\ x_{4}^{j}&x_{1}^{j}&x_{2}^{j}&x_{3}^{j}\end{pmatrix}$ for $j=1,2,4,5,6,7,8$;

$\begin{pmatrix}x_{i}^{1}&x_{i}^{2}&x_{i}^{3}&x_{i}^{4}\\ x_{i}^{7}&x_{i}^{5}&x_{i}^{6}&x_{i}^{8}\end{pmatrix}$ for $i=2,4,6$; $\begin{pmatrix}x_{1}^{j}&x_{8}^{j}&x_{2}^{j}&x_{9}^{j}\\ x_{5}^{j}&x_{6}^{j}&x_{4}^{j}&x_{7}^{j}\end{pmatrix}$ for $j=1,2,3,4,7,8,9$;

$\begin{pmatrix}x_{i}^{1}&x_{i}^{2}&x_{i}^{3}&x_{i}^{4}\\ x_{i}^{7}&x_{i}^{6}&x_{k}^{3}&x_{i}^{8}\end{pmatrix}$ for $(i,k)=(1,7),(7,2)$; $\begin{pmatrix}x_{i}^{1}&x_{i}^{2}&x_{i}^{3}&x_{i}^{4}\\ x_{i}^{7}&x_{k}^{2}&x_{i}^{6}&x_{i}^{8}\end{pmatrix}$ for $(i,k)=(3,5),(8,4)$;

$\begin{pmatrix}x_{i}^{1}&x_{i}^{2}&x_{i}^{3}&x_{i}^{4}\\ x_{k}^{1}&x_{i}^{5}&x_{i}^{6}&x_{k}^{4}\end{pmatrix}$ for $(i,k)=(5,3),(9,6)$; $\begin{pmatrix}x_{i}^{5}&x_{i}^{6}&x_{i}^{7}&x_{i}^{8}\\ x_{i}^{4}&x_{i}^{1}&x_{i}^{2}&x_{i}^{3}\end{pmatrix}$ for $i=1,3,4,6,7,8$;

$\begin{pmatrix}x_{i}^{1}&x_{i}^{8}&x_{i}^{2}&x_{i}^{9}\\ x_{i}^{5}&x_{i}^{6}&x_{i}^{4}&x_{i}^{7}\end{pmatrix}$ for $i=1,3,\dots,9$; $\begin{pmatrix}x_{i}^{3}&x_{i}^{7}&x_{i}^{4}&x_{i}^{9}\\ x_{i}^{1}&x_{i}^{5}&x_{i}^{6}&x_{i}^{8}\end{pmatrix}$ for $i=1,3,4,5,7,8,9$;

$\begin{pmatrix}x_{3}^{j}&x_{7}^{j}&x_{4}^{j}&x_{9}^{j}\\ x_{1}^{j}&x_{5}^{j}&x_{6}^{j}&x_{8}^{j}\end{pmatrix}$ for $j=1,2,3,4,6,7,8,9$;

$\begin{pmatrix}x_{1}^{1}&x_{2}^{1}&x_{3}^{1}&x_{4}^{1}\\ x_{7}^{1}&x_{2}^{5}&x_{6}^{1}&x_{8}^{1}\end{pmatrix}$, $\begin{pmatrix}x_{1}^{2}&x_{2}^{2}&x_{3}^{2}&x_{4}^{2}\\ x_{7}^{2}&x_{5}^{2}&x_{6}^{2}&x_{4}^{6}\end{pmatrix}$, $\begin{pmatrix}x_{1}^{3}&x_{2}^{3}&x_{3}^{3}&x_{4}^{3}\\ x_{1}^{6}&x_{2}^{8}&x_{6}^{3}&x_{8}^{3}\end{pmatrix}$, $\begin{pmatrix}x_{1}^{5}&x_{2}^{5}&x_{3}^{5}&x_{4}^{5}\\ x_{1}^{3}&x_{2}^{7}&x_{6}^{5}&x_{4}^{3}\end{pmatrix}$,

$\begin{pmatrix}x_{1}^{9}&x_{2}^{9}&x_{3}^{9}&x_{4}^{9}\\ x_{1}^{6}&x_{5}^{9}&x_{6}^{9}&x_{4}^{6}\end{pmatrix}$, $\begin{pmatrix}x_{5}^{3}&x_{6}^{3}&x_{7}^{3}&x_{8}^{3}\\ x_{4}^{3}&x_{1}^{3}&x_{7}^{6}&x_{8}^{5}\end{pmatrix}$, $\begin{pmatrix}x_{5}^{9}&x_{6}^{9}&x_{7}^{9}&x_{8}^{9}\\ x_{5}^{5}&x_{9}^{9}&x_{2}^{9}&x_{3}^{9}\end{pmatrix}$, $\begin{pmatrix}x_{1}^{5}&x_{8}^{5}&x_{2}^{5}&x_{9}^{5}\\ x_{5}^{5}&x_{6}^{5}&x_{2}^{3}&x_{7}^{5}\end{pmatrix}$,

$\begin{pmatrix}x_{1}^{6}&x_{8}^{6}&x_{2}^{6}&x_{9}^{6}\\ x_{5}^{6}&x_{8}^{2}&x_{4}^{6}&x_{7}^{6}\end{pmatrix}$, $\begin{pmatrix}x_{3}^{5}&x_{7}^{5}&x_{4}^{5}&x_{9}^{5}\\ x_{1}^{5}&x_{7}^{2}&x_{6}^{5}&x_{8}^{5}\end{pmatrix}$, $\begin{pmatrix}x_{2}^{5}&x_{2}^{6}&x_{2}^{7}&x_{2}^{8}\\ x_{2}^{4}&x_{2}^{1}&x_{2}^{2}&x_{6}^{8}\end{pmatrix}$, $\begin{pmatrix}x_{5}^{5}&x_{5}^{6}&x_{5}^{7}&x_{5}^{8}\\ x_{7}^{5}&x_{5}^{1}&x_{5}^{2}&x_{5}^{3}\end{pmatrix}$,

$\begin{pmatrix}x_{9}^{5}&x_{9}^{6}&x_{9}^{7}&x_{9}^{8}\\ x_{5}^{5}&x_{9}^{9}&x_{9}^{2}&x_{9}^{3}\end{pmatrix}$, $\begin{pmatrix}x_{2}^{1}&x_{2}^{8}&x_{2}^{2}&x_{2}^{9}\\ x_{6}^{1}&x_{2}^{6}&x_{2}^{4}&x_{2}^{7}\end{pmatrix}$, $\begin{pmatrix}x_{2}^{3}&x_{2}^{7}&x_{2}^{4}&x_{2}^{9}\\ x_{2}^{1}&x_{6}^{7}&x_{2}^{6}&x_{2}^{8}\end{pmatrix}$, $\begin{pmatrix}x_{6}^{3}&x_{6}^{7}&x_{6}^{4}&x_{6}^{9}\\ x_{6}^{1}&x_{6}^{5}&x_{2}^{4}&x_{6}^{8}\end{pmatrix}$,

$\begin{pmatrix}x_{6}^{2}&x_{6}^{6}&x_{9}^{6}&x_{9}^{2}\\ x_{2}^{2}&x_{8}^{6}&x_{5}^{6}&x_{5}^{2}\end{pmatrix}$, $\begin{pmatrix}x_{6}^{3}&x_{6}^{5}&x_{9}^{5}&x_{9}^{3}\\ x_{2}^{3}&x_{2}^{5}&x_{9}^{9}&x_{5}^{3}\end{pmatrix}$, $\begin{pmatrix}x_{5}^{4}&x_{5}^{8}&x_{9}^{8}&x_{9}^{4}\\ x_{5}^{5}&x_{3}^{8}&x_{6}^{8}&x_{9}^{5}\end{pmatrix}$, $\begin{pmatrix}x_{5}^{1}&x_{5}^{7}&x_{9}^{7}&x_{9}^{1}\\ x_{2}^{1}&x_{3}^{7}&x_{6}^{7}&x_{9}^{6}\end{pmatrix}$,

$\begin{pmatrix}x_{3}^{6}&x_{3}^{9}&x_{5}^{9}&x_{5}^{6}\\ x_{3}^{2}&x_{3}^{5}&x_{9}^{9}&x_{5}^{2}\end{pmatrix}$, $\begin{pmatrix}x_{3}^{3}&x_{3}^{5}&x_{5}^{5}&x_{5}^{3}\\ x_{8}^{3}&x_{3}^{2}&x_{2}^{5}&x_{2}^{3}\end{pmatrix}$, $\begin{pmatrix}x_{2}^{6}&x_{2}^{9}&x_{6}^{9}&x_{6}^{6}\\ x_{2}^{2}&x_{2}^{5}&x_{6}^{5}&x_{6}^{4}\end{pmatrix}$, $\begin{pmatrix}x_{1}^{5}&x_{1}^{9}&x_{7}^{9}&x_{7}^{5}\\ x_{1}^{2}&x_{6}^{9}&x_{7}^{6}&x_{7}^{3}\end{pmatrix}$,

$\begin{pmatrix}x_{4}^{5}&x_{4}^{9}&x_{8}^{9}&x_{8}^{5}\\ x_{2}^{5}&x_{5}^{9}&x_{8}^{6}&x_{8}^{2}\end{pmatrix}$.