Isometry invariant permutation codes and mutually orthogonal Latin squares

TL;DR

This paper constructs new sets of mutually orthogonal Latin squares (MOLS) using isometry invariant permutation codes, providing improved lower bounds for the number of MOLS for specific orders and offering a computational approach for their description.

Contribution

It introduces a novel method to construct MOLS via isometry invariant permutation codes, resulting in new lower bounds for certain orders and a uniform computational description.

Findings

Constructed permutation codes of lengths 35, 48, 63, 96 with specified minimum distances.

Derived new lower bounds for the number of MOLS for n=35, 48, 63, 96.

Codes are described using subgroup generators of the isometry group, facilitating computational analysis.

Abstract

Commonly the direct construction and the description of mutually orthogonal Latin squares (MOLS) makes use of difference or quasi-difference matrices. Now there exists a correspondence between MOLS and separable permutation codes. We like to present separable permutation codes of length $35$, $48$, $63$ and $96$ and minimum distance $34$, $47$, $62$ and $95$ consisting of $6 \times 35$, $10 \times 48$, $8 \times 63$ and $8 \times 96$ codewords respectively. Using the correspondence this gives $6$ MOLS for $n=35$, $10$ MOLS for $n=48$, $8$ MOLS for $n=63$ and $8$ MOLS for $n=96$. So $N(35) \ge 6$, $N(48) \ge 10$, $N(63) \ge 8$ and $N(96) \ge 8$ holds which are new lower bounds for MOLS. The codes will be given by generators of an appropriate subgroup $U$ of the isometry group of the symmetric group $S_n$ and $U$-orbit representatives. This gives an alternative uniform way to describe the MOLS where the data for the codes can be used as input for computer algebra systems like MAGMA, GAP etc.

Equations

$$d(C):=\min \{d_H(\sigma ,\tau )\mid \sigma ,\tau \in C,\sigma \ne \tau \}.$$

$$b*x:=\left\{\begin{array}[]{lll}x^{-1}b&\mbox{if}&x\in B_{1}\\ b\varphi(x)&\mbox{if}&x\in B_{2}\\ b^{-1}&\mbox{if}&x=t_{n},\\ \end{array}\right.$$

$$d_{ik}^{{\gamma }_j}=d_{ik}^{{\theta }_i^{-1}{\theta }_{i^{\prime }}}={\left(d_{ik}^{{\theta }_i^{-1}}\right)}^{{\theta }_{i^{\prime }}}=g_k^{{\theta }_{i^{\prime }}}\text{mbox}and{d}_{i\ell }^{{\gamma }_j}=d_{i\ell }^{{\theta }_i^{-1}{\theta }_{i^{\prime }}}={\left(d_{i\ell }^{{\theta }_i^{-1}}\right)}^{{\theta }_{i^{\prime }}}=g_{\ell }^{{\theta }_{i^{\prime }}}.$$

$$d_{i^{\prime }k}=g_k^{{\theta }_{i^{\prime }}}=d_{ik}^{{\gamma }_j}=d_{ik}{g}_j\text{mbox}and{d}_{i^{\prime }\ell }=g_{\ell }^{{\theta }_{i^{\prime }}}=d_{i\ell }^{{\gamma }_j}=d_{i\ell }{g}_j.$$

$$d_{ik}^{-1}{d}_{i^{\prime }k}=d_{ik}^{-1}(d_{ik}{g}_j)=g_j=d_{i\ell }^{-1}(d_{i\ell }{g}_j)=d_{i\ell }^{-1}{d}_{i^{\prime }\ell },$$

$$g_j:={\left(g_k^{{\theta }_i}\right)}^{-1}{g}_k^{{\theta }_{i^{\prime }}}={\left(g_{\ell }^{{\theta }_i}\right)}^{-1}{g}_{\ell }^{{\theta }_{i^{\prime }}},g_{k^{\prime }}=g_k^{{\theta }_i},g_{{\ell }^{\prime }}=g_{\ell }^{{\theta }_i}.$$

$$g_{k^{\prime }}^{{\gamma }_j{\theta }_{i^{\prime }}^{-1}}={\left({\left(g_k^{{\theta }_i}\right)}^{{\gamma }_j}\right)}^{{\theta }_{i^{\prime }}^{-1}}={\left(g_k^{{\theta }_i}{g}_j\right)}^{{\theta }_{i^{\prime }}^{-1}}=$$

$${\left(g_k^{{\theta }_i}\left({\left(g_k^{{\theta }_i}\right)}^{-1}{g}_k^{{\theta }_{i^{\prime }}}\right)\right)}^{{\theta }_{i^{\prime }}^{-1}}={\left(g_k^{{\theta }_{i^{\prime }}}\right)}^{{\theta }_{i^{\prime }}^{-1}}=g_k=g_{k^{\prime }}^{{\theta }_i^{-1}}$$

$$g_{{\ell }^{\prime }}^{{\gamma }_j{\theta }_{i^{\prime }}^{-1}}=g_{\ell }=g_{{\ell }^{\prime }}^{{\theta }_i^{-1}}.$$

Full text

K****eywords bounds, isometry, permutation code, permutation arrays, MOLS, mutually orthogonal Latin squares

1 Introduction

Let $n$ be an integer and $V$ and $W$ be sets consisting of $n$ elements. A Graeco-Latin square is a $n\times n$ matrix $M=((v_{ij},w_{ij}))$ with entries in the Cartesian product $V\times W$ such that the set of all different entries in $M$ equals $V\times W$ and the matrices $L_{1}:=(v_{ij})$ and $L_{2}:=(w_{ij})$ are Latin squares, i.e. all rows and all columns of $L_{1}$ and $L_{2}$ are permutations of $V$ and $W$ respectively. On the other hand two such Latin squares are called orthogonal if the matrix $((v_{ij},w_{ij}))$ is a Graeco-Latin square. A set of mutually orthogonal Latin squares (MOLS) consists of Latin squares which are pairwise orthogonal. Let $N(n)$ be the largest size for MOLS of order $n$. In 1960 R.C. Bose, S.S Shrikhande and E.T. Parker [3] proved the existence of a Graeco-Latin square (which is equivalent to $N(n)\geq 2$) for all integers $n\geq 3$, $n\neq 6$ which disproved a conjecture from L. Euler in 1782 [7]. It is easy to that $N(n)\leq n-1$ and equality holds if $n$ is a prime power but for all other integers $n\geq 10$ the number $N(n)$ is not known and there just exist lower bounds.

In the last decades improving a lower bound for $N(n)$ was always obtained either by construction of a $(G,m+1;\lambda)$-difference matrix a $(G,m+2;\lambda,\mu;u)$-quasi-difference matrix for a finite group $G$ or recursive constructions. For the definitions of these kind of matrices see $17.1$ and $17.44$ in [5].

In this paper we like to proceed in a different way. For explanation let us recall the notation in [9] and some definitions from there.

Let $n$ be a positive integer and let $X$ be an arbitrary finite set of order $n$. Let $S_{X}:=\{\sigma:X\rightarrow X\mid\sigma\;\mbox{is bijective}\}.$ This group acts on $X$ from right and for $x\in X,\sigma\in S_{X},$ we denote the image of $x$ under $\sigma$ by $x^{\sigma}.$ If $X=\{1,2,\ldots,n\}$ we will write $S_{n}$ instead of $S_{X}.$ For $\sigma,\tau\in S_{X}$ the Hamming distance $d_{H}(\sigma,\tau)=d(\sigma,\tau)$ equals $n$ minus the number of fixed points of $\sigma^{-1}\tau$, and for $S,T\subseteq S_{X}$ let $d(S,T):=\min\ \{d(\sigma,\tau)\mid\sigma\in S,\tau\in T\}.$ A subset $C$ of $S_{X}$ is called a permutation code or permutation array of length $n$ and of minimum distance

[TABLE]

For brevity such a permutation code $C$ is called an $(n,d)$-PA. An $(n,n-1)$-PA $C$ is called $(r,m)$-separable if it is the join of $m$ disjoint $(n,n)$-PAs $L_{1},\ldots,L_{m}$ of cardinality $r$ such that for all pairs $L_{i},L_{j}$, $i\neq j$ the distance $d_{H}(\sigma,\tau)$ equals $n-1$ for all $\sigma\in L_{i}$ and all $\tau\in L_{j}$. For $r=n$ the code $C$ corresponds to $m$ MOLS [6].

$S_{n}$ is a metric space via the Hamming distance and the isometry group $\mathit{Iso}(n)$ is isomorphic to the wreath product $S_{n}\wr S_{2}$ [8]. $\mathit{Iso}(n)$ can be described as subgroup of $S_{2n}$. Let $B_{1}$ and $B_{2}$ be the naturally embedded subgroups isomorphic to $S_{n}$ acting on the sets $\{1,2,\ldots,n\}$ and $\{n+1,n+2,\ldots,2n\}$, respectively. Moreover let $t_{n}:=(1,n+1)(2,n+2)\cdots(n,2n)\in S_{2n}$. Then $\mathit{Iso}(n)=\langle B_{1},B_{2},t_{n}\rangle=(B_{1}\times B_{2}):\langle t_{n}\rangle$, and the action of this group on $B_{1}$ from right is given by

[TABLE]

where $b\in B_{1}$, $x\in Iso(n)$ and $\varphi$ denotes the natural isomorphism from $B_{2}$ to $B_{1}$. Moreover, if $U$ is a subgroup of $\mathit{Iso}(n)$ and $b\in B_{1},$ $b*U=\{b*u\mid u\in U\}$ denotes the $U-$orbit of $b.$ Our codes $C$ will now be regarded as subsets of $B_{1}$ and a code $C$ will be called $U-$invariant if $C$ is closed under the action of $U$, which means that $C$ is a union of $U-$orbits. The strategy of constructing $(n,m)$-separable PAs invariant under a given subgroup $U$ of $\mathit{Iso}(n)$ we like to refer to [9]. But now we will join only separable $U$-orbits.

In section 3 we will proof the existence of $6$ MOLS for $n=35$, $10$ MOLS for $n=48$, $8$ MOLS for $n=63$ and $n=96$ by constructing a $(35,6)$ - separable, a $(48,10)$ - separable, a $(63,8)$ - separable and a $(96,8)$ - separable PA respectively. These are invariant under specific subgroups $U$ of $Iso(n)$ using the described group action of $Iso(n)$ on $S_{n}$ and will be given by generators of $U$ and representatives of the $U$-orbits.

Furthermore we like to introduce a correspondence between $(n,m+2,1)$-difference matrices and $(n,m)$-separable PAs with special kind of isometry groups in section 2

In our studies we also investigated cases different from $n=35$, $n=48$, $n=63$ and $n=96.$ Some interesting results are summarized in section 4

Since all data for the constructed PAs are contained in the symmetric group $S_{2n}$ it can easily be read into a computer algebra system like MAGMA [4] or GAP [12]. This is a uniform and compact way to describe $m$ MOLS of order $n$ with natural algebraic objects.

2 Difference matrices and cosets

In this section we will prove that the concept of difference matrices over a group $G$ agrees with the concept of separable permutation codes having an isometry group which contains a special group which is naturally isomorphic to $G.$

Let $G=\{g_{1}=1_{G},g_{2},\ldots g_{n}\}.$ For each $j\in\{1,2,\ldots,n\}$ let $\gamma_{j}:G\rightarrow G$ be the bijection defined by $g^{\gamma_{j}}:=gg_{j}.$ Now $g^{\gamma_{i}\gamma_{j}}=\left(g^{\gamma_{i}}\right)^{\gamma_{j}}=(gg_{i})g_{j}=g(g_{i}g_{j})$ shows that mapping $g_{i}$ to $\gamma_{i}$ is an isomorphism between $G$ and ${\cal{R}}(G):=\{\gamma_{1}=Id,\gamma_{2},\ldots\gamma_{n}\}\leq S_{G}.$ ${\cal{R}}(G)$ is called the (right-)regular representation of $G$.

Proposition 1**.**

Let $G$ be a finite group of order $n$. A $(G,m+1;1)-$difference matrix D exists if and only if there exists a set $\{\theta_{1}=Id,\theta_{2},\ldots,\theta_{m}\}\subseteq S_{G}$ such that $d({\cal{R}}(G)\theta_{i^{\prime}},\theta_{i})=n-1$ for all $i,i^{\prime}\in\{1,2,\ldots,m\}$ with $i\neq i^{\prime}.$

Proof.

Let $D=(d_{ik})$ be a $(G,m+1;1)$-difference matrix, where we number the rows from [math] to $m$. Then $\mid\{d_{ik}^{-1}d_{jk}\mid k\in\{1,2,\ldots,n\}\}\mid=n$ for all $i,j\in\{0,1,\ldots m\}$ with $i\neq j$. We may assume that $D$ is normalized, i.e. all elements of the zero-th row are equal to $g_{1}=1_{G}$ and $d_{1k}=g_{k}$ for all $k\in\{1,2,\ldots,n\}.$ For each $i\in\{1,2,\ldots m\}$ we define $\theta_{i}:G\rightarrow G$ by $g_{k}^{\theta_{i}}=d_{ik}$. Then $\theta_{i}\in S_{G}$ for any $i\in\{1,2,\ldots m\}$ and $\theta_{1}=Id$.

$d({\cal{R}}(G)\theta_{i^{\prime}}^{-1},\theta_{i}^{-1})\neq n$ since $\mid{\cal{R}}(G)\mid=d({\cal{R}}(G))=n$. To show $d({\cal{R}}(G)\theta_{i^{\prime}}^{-1},\theta_{i}^{-1})=n-1$ for all $i,i^{\prime}\in\{1,2,\ldots,m\}$ with $i\neq i^{\prime}$ let us assume the contrary, i.e. there exist $\gamma_{j}\in{\cal{R}}(G),g_{k^{\prime}},g_{\ell^{\prime}}\in G$ with $k^{\prime}\neq\ell^{\prime}$ and $g_{k^{\prime}}^{\gamma_{j}\theta_{i^{\prime}}^{-1}}=g_{k^{\prime}}^{\theta_{i}^{-1}}$ and $g_{\ell^{\prime}}^{\gamma_{j}\theta_{i^{\prime}}^{-1}}=g_{\ell^{\prime}}^{\theta_{i}^{-1}},$ which implies $g_{k^{\prime}}^{\gamma_{j}}=g_{k^{\prime}}^{\theta_{i}^{-1}\theta_{i^{\prime}}}$ and $g_{\ell^{\prime}}^{\gamma_{j}}=g_{\ell^{\prime}}^{\theta_{i}^{-1}\theta_{i^{\prime}}}.$ Then there exist $k,\ell\in\{1,2,\ldots,n\}$ with $k\neq l$ such that $d_{ik}=g_{k^{\prime}}$ and $d_{i\ell}=g_{\ell^{\prime}}$ so that

[TABLE]

We conclude

[TABLE]

Thus

[TABLE]

which contradicts the properties of a difference matrix.

Now we like to show the other direction. Again we number the rows from [math] to $m$ and define $D$ as follows: The elements of the $0-$th row are all equal to $1_{G}.$ For all $i\in\{1,2,\ldots,m\}$ and for all $k\in\{1,2,\ldots,n\}$ let $d_{ik}:=g_{k}^{\theta_{i}}$. We claim that $D$ is a difference matrix. For all $i\in\{1,2,\ldots,m\}$ the elements in the i-th row are different since $\theta_{i}$ is a bijection. Thus we only need to show that for $i,i^{\prime}\in\{1,2,\ldots,m\}$ with $i\neq i^{\prime}$ all the elements $d_{i^{\prime}k}^{-1}d_{ik}$ are pairwise different for $k\in\{1,2,\ldots,n\}$.

Suppose that $k,\ell\in\{1,2,\ldots,n\}$, $k\neq\ell$ with $d_{ik}^{-1}d_{i^{\prime}k}=d_{i\ell}^{-1}d_{i^{\prime}\ell}.$ Let

[TABLE]

This implies $k^{\prime}\neq\ell^{\prime}$ and

[TABLE]

[TABLE]

and similarly

[TABLE]

Hence the maps $\gamma_{j}\theta_{i^{\prime}}^{-1}$ and $\theta_{i}^{-1}$ have the same values on $g_{k^{\prime}}$ and on $g_{\ell^{\prime}}$ which means that $d\left(\gamma_{j}\theta_{i^{\prime}}^{-1},\theta_{i}^{-1}\right)<n-1,$ which contradicts the assumption. ∎

Let $H$ be a group of order less or equal to $n$ and let $\Phi\ :S_{H}\rightarrow B_{1}\leq Iso(n)$ be the natural embedding. Then proposition 1 implies

Theorem 1**.**

Let $G$ be a group of order $n$. There is a one to one correspondence between the normalized $(G,m+1;1)-$difference matrices and $\Phi({\cal R}(G))-$invariant codes of length $m\cdot n$ with minimum distance $n-1$ containing the set $\Phi({\cal R}(G))$.

Hence our concept of choosing a subgroup $U$ of $\mathit{Iso}(n)$ and calculating all separable $U-$orbits with minimum distance at least $n-1$ and finding a subset of these orbits such that the union is an $(n,m)-$separable PA with $m$ as large as possible covers the search for MOLS via difference matrices. In particular the proof shows a way how to construct a separable $(n,m)$ - PA from a $(G,m+1;1)-$difference matrix and vice versa.

Proposition 1 shows that a normalized $(G,m+1;1)-$difference matrix corresponds to $m$ cosets of the right regular representation of $G$ (including the trivial one), where two different cosets have minimum distance equal to $n-1$ from each other. The natural question arises if something similar is true for $(G,m+1;\lambda)-$difference matrices with $\lambda>1$ and for $(G,m+2;\lambda,\mu;u)$-quasi-difference matrices, respectively.

In the cases for $n\in\{22,26,30,34\}$ we converted $(G,m+2;\lambda,\mu;u)$-quasi-difference matrices from [5]. It turned out that the $(n,m)$ - separable PAs obtained this way have an isometry group which contains the subgroup $\Delta(\Phi({\cal R}(G)))$ of order $n-u$ where $\Delta(V):=\{vv^{t_{n}}\mid v\in V\}$ for a subgroup $V$ of $B_{1}$.

Moreover, each individual $(n,n)-$PA of size $n$ contained in the $(n,m)-$ separable PA always consists of one orbit of size $n-u$ (a regular orbit) and of $u$ many orbits of size $1$.

Concerning the case of $(G,m+1;\lambda)-$difference matrices with $\lambda>1,$ there are only two cases where the best known bound for $N(n)$ is realized in this way namely $n=14$ and $n=18$ ( [13] resp. [2] ) where $\lambda=2$ in both cases. We converted the difference matrices from [13] and [2] to a $(14,4)$ - separable PA and a $(18,5)$ - separable PA respectively. It turned out that $\Delta(U)$ is an isometry group of the corresponding PA where $U$ is the unique subgroup of index $2$ in $\Phi({\cal R}({\mathbb{Z}}_{14}))$ resp. $\Phi({\cal R}({\mathbb{Z}}_{3}\times{\mathbb{Z}}_{6}))$. In fact there are bigger isometry groups containing these groups.

Maybe one can generalize these observations.

3 Improvements

Let $n=35$. In [15] M. Wojtas proved the existence of 5 MOLS by constructing a $(35,6,1)$ difference matrix. This implies $N(35)\geq 5$. In the following we will improve this bound. Let $E_{35}:=\langle x\rangle=\Phi({\cal R}({\mathbb{Z}}_{35}))\leq B_{1}$ of order $35$ generated by $x$ and let $U:=\langle\Delta(E_{35}),y_{1},y_{2}\rangle\leq Iso(35)$ generated by $\Delta(E_{35})$ and $y_{1},y_{2}$ where

$x:=(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,\\ \ 27,28,29,30,31,32,33,34,35),\\ y_{1}:=(1,36)(2,37)(3,38)(4,39)(5,40)(6,41)(7,42)(8,43)(9,44)(10,45)(11,46)(12,47)(13,48)\\ \ (14,49)(15,50)(16,51)(17,52)(18,53)(19,54)(20,55)(21,56)(22,57)(23,58)(24,59)(25,60)\\ \ (26,61)(27,62)(28,63)(29,64)(30,65)(31,66)(32,67)(33,68)(34,69)(35,70),\\ y_{2}:=(1,58)(2,57)(3,56)(4,55)(5,54)(6,53)(7,52)(8,51)(9,50)(10,49)(11,48)(12,47)(13,46)\\ \ (14,45)(15,44)(16,43)(17,42)(18,41)(19,40)(20,39)(21,38)(22,37)(23,36)(24,70)(25,69)\\ \ (26,68)(27,67)(28,66)(29,65)(30,64)(31,63)(32,62)(33,61)(34,60)(35,59).$

Furthermore let

$a:=(2,18,23,25)(3,21,24,28)(4,31,7,33,6,30)(5,20,26,27)(8,22,29,15)(9,34,16,13)\\ \ (10,32,17,11)(12,35,19,14),\\ b:=(1,32,13,7,34,24,15,2,5,19,8)(3,33,10,28,29,31,16,21,9,26,25,23)\\ \ (4,17,14,35,11,18,22,6,12,20,30),\\ c:=(1,25)(2,24)(3,23)(4,22)(5,21)(6,20)(7,19)(8,18)(9,17)(10,16)(11,15)(12,14)(26,35)\\ \ (27,34)(28,33)(29,32)(30,31),\\$

$a,b,c$ are codewords of $B_{1}$. The orbits $a*U$, $b*U$, $c*U$ have cardinality $70$, $70$, $35$ respectively and $E_{35}$ splits into $18$ $U$ - orbits of lengths $1$ and $2$. The $(35,34)$ - PA $C:=a*U\cup b*U\cup c*U\cup E_{35}$ is $(35,6)$ - separable. This proves

Theorem 2**.**

$N(35)\geq 6$.

The isometry group $U$ is an extensions from $\Delta(E_{35})$ of index $4$. We calculated the separable orbits stabilized by a subgroup of order $2$. A backtrack search gave our result.

Let $n=48$. In [1] Abel and Cavenagh proved the existence of 8 MOLS by constructing a $(48,9,1)$ difference matrix. This implies $N(48)\geq 8$. Now we like to improve this bound. Let $E_{48}:=\langle x_{1},x_{2},x_{3},x_{4}\rangle=\Phi({\cal R}({\mathbb{Z}}_{6}\times{\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}))\leq B_{1}$ of order $48$ generated by $x_{1},x_{2},x_{3},x_{4}$ and let $U:=\langle\Delta(E_{48}),y_{1},y_{2}\rangle\leq Iso(48)$ of order $1152$ generated by $\Delta(E_{48})$ and $y_{1},y_{2}$ where

$x_{1}:=(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)\\ \ (27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)(41,42)(43,44)(45,46)(47,48),\\ x_{2}:=(1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27)\\ \ (26,28)(29,31)(30,32)(33,35)(34,36)(37,39)(38,40)(41,43)(42,44)(45,47)(46,48),\\ x_{3}:=(1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)\\ \ (26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(41,45)(42,46)(43,47)(44,48),\\ x_{4}:=(1,25,33,9,17,41)(2,26,34,10,18,42)(3,27,35,11,19,43)(4,28,36,12,20,44)\\ \ (5,29,37,13,21,45)(6,30,38,14,22,46)(7,31,39,15,23,47)(8,32,40,16,24,48),\\ y_{1}:=(1,18,36)(2,20,33)(3,19,35)(4,17,34)(5,32,42)(6,30,43)(7,29,41)(8,31,44)(9,23,45)\\ \ (10,21,48)(11,22,46)(12,24,47)(13,25,39)(14,27,38)(15,28,40)(16,26,37)(49,84,67)\\ \ (50,82,66)(51,81,68)(52,83,65)(53,94,73)(54,96,76)(55,95,74)(56,93,75)(57,85,78)\\ (58,87,79)(59,88,77)(60,86,80)(61,91,72)(62,89,69)(63,90,71)(64,92,70),\\ y_{2}:=(1,72,10,77)(2,69,9,80)(3,70,12,79)(4,71,11,78)(5,74,14,67)(6,75,13,66)\\ \ (7,76,16,65)(8,73,15,68)(17,88,26,93)(18,85,25,96)(19,86,28,95)(20,87,27,94)\\ \ (21,90,30,83)(22,91,29,82)(23,92,32,81)(24,89,31,84)(33,56,42,61)(34,53,41,64)\\ \ (35,54,44,63)(36,55,43,62)(37,58,46,51)(38,59,45,50)(39,60,48,49)(40,57,47,52).$

Furthermore let

$a:=(1,38,2,33,12,44,14,6,40,26,28,41,39,43)(3,30,29,25,32,48,11,47,8,15,42,\\ \ 22,37,13,18,36,31,5,35,46,19,21,16,45,34,7,27,4,10)(9,23,20,17),\\ b:=(2,4,45,21,17,7,23,44,26,15,30,25,14,5,13,8,22,20,39,43,27,47,34,11,37,\\ \ 32,36,19,38,29,28,46,24,41,3,48,35,18,6,16,31,33,10,9,12,40,42).$

$a,b$ are codewords of $B_{1}$. The orbits $a*U$ and $b*U$ have cardinality $288$ resp. $144$ and $E_{48}$ splits into one $U$ - orbit of length $24$ and two orbits of lengths $12$. The $(48,47)$ - PA $C:=a*U\cup b*U\cup E_{48}$ is $(48,10)$ - separable. This proves

Theorem 3**.**

$N(48)\geq 10$.

The isometry group $U$ is an extensions from $\Delta(E_{48})$ of index $24$. We calculated the separable orbits stabilized by a subgroup of order $4$. A backtrack search gave our result.

In 1922 McNeish proved his well-known bound on $N(n)$ [10]. For all $n<63$ there exist improvements for this bound [5], [2]. Up to now the best known bound for $n=63$ is McNeish’s bound $N(63)\geq 6$. Thus $n=63$ is of special interest. We are now able to construct a $(63,8)$ - separable PA. Let $E_{63}:=\langle x_{1},x_{2}\rangle=\Phi({\cal R}({\mathbb{Z}}_{3}\times{\mathbb{Z}}_{21}))\leq B_{1}$ of order $63$ generated by $x_{1},x_{2}$ and let $U:=\langle\Delta(E_{63}),y_{1},y_{2}\rangle\leq Iso(63)$ of order $3402$ generated by $\Delta(E_{63})$ and $y_{1},y_{2}$ where

$x_{1}:=(1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24)(25,26,27)\\ \ (28,29,30)(31,32,33)(34,35,36)(37,38,39)(40,41,42)(43,44,45)(46,47,48)(49,50,51)\\ \ (52,53,54)(55,56,57)(58,59,60)(61,62,63),\\ x_{2}:=(1,13,25,28,40,52,55,4,16,19,31,43,46,58,7,10,22,34,37,49,61)(2,14,26,29,\\ \ 41,53,56,5,17,20,32,44,47,59,8,11,23,35,38,50,62)(3,15,27,30,42,54,57,6,\\ \ 18,21,33,45,48,60,9,12,24,36,39,51,63),\\ y_{1}:=(1,42,56)(2,37,60)(3,44,61)(4,45,59)(5,40,63)(6,38,55)(7,39,62)(8,43,57)(9,41,58)(10,15,11)\\ \ (12,17,16)(13,18,14)(19,51,29)(20,46,33)(21,53,34)(22,54,32)(23,49,36)(24,47,28)(25,48,35)\\ \ (26,52,30)(27,50,31)(64,108,125)(65,103,120)(66,101,121)(67,102,119)(68,106,123)(69,104,124)\\ \ (70,105,122)(71,100,126)(72,107,118)(73,81,80)(74,76,75)(77,79,78)(82,117,98)(83,112,93)\\ \ (84,110,94)(85,111,92)(86,115,96)(87,113,97)(88,114,95)(89,109,99)(90,116,91),\\ y_{2}:=(1,79,50,69,18,110)(2,81,51,68,16,109)(3,80,49,67,17,111)(4,76,53,66,12,116)\\ \ (5,78,54,65,10,115)(6,77,52,64,11,117)(7,73,47,72,15,113)(8,75,48,71,13,112)(9,74,46,70,14,114)\\ \ (19,88,23,87,27,83)(20,90,24,86,25,82)(21,89,22,85,26,84)(28,124,41,96,63,101)\\ \ (29,126,42,95,61,100)(30,125,40,94,62,102)(31,121,44,93,57,107)(32,123,45,92,55,106)\\ \ (33,122,43,91,56,108)(34,118,38,99,60,104)(35,120,39,98,58,103)(36,119,37,97,59,105).$

Furthermore let

$a:=(2,49,26,29,47,22,28,53,27,32,54,39,41,36,52,24,7,51,23,31,60,61,62,20,9,3,16,6,50,38,11,\\ \ 4,17,56,63,14,45,34,46,37,35,58,19,33,48,25,8,18,43,10,44,40,42,12,57,21,30,59,13,55,15,5),\\ b:=(2,3)(4,7)(5,9)(6,8)(10,47,17,46,15,48)(11,52,18,54,13,53)(12,51,16,50,14,49)(19,32,20,28,21,36)\\ \ (22,29,23,34,24,33)(25,35,26,31,27,30)(37,62,41,55,45,60)(38,58,42,63,43,56)(39,57,40,59,44,61).$

$a,b$ are codewords of $B_{1}$. The orbits $a*U$ and $b*U$ have cardinality $378$ resp. $63$. and $E_{63}$ splits into $U$ - orbits of length $54$ and $9$. The $(63,62)$ - PA $C:=a*U\cup b*U\cup E_{63}$ is $(63,8)$ - separable. This proves

Theorem 4**.**

$N(63)\geq 8$.

The isometry group $U$ is an extensions from $\Delta(E_{63})$ of index $54$. We calculated the separable orbits stabilized by a subgroup of order $9$. A backtrack search gave our result.

We also found 7 MOLS using an extension of $\Phi({\cal R}({\mathbb{Z}}_{3}\times{\mathbb{Z}}_{21}))$ of index $18$ which therefore corresponds to a difference matrix by theorem 1. Notice that $n=72$ now is the smallest $n$ for which MacNeish’s theorem gives the best known lower bound for $N(n)$.

Let now $n=96$ and let $U:=\langle x,y,z\rangle\leq Iso(96)$ of order $4608$ generated by $x,y,z$ given in cycle structure

$x:=(1,30,33,14,17,46)(2,29,34,13,18,45)(3,23,35,7,19,39)(4,24,36,8,20,40)\\ \ (5,26,37,10,21,42)(6,25,38,9,22,41)(11,31,43,15,27,47)(12,32,44,16,28,48)\\ \ (49,78,81,62,65,94)(50,77,82,61,66,93)(51,71,83,55,67,87)(52,72,84,56,68,88)\\ \ (53,74,85,58,69,90)(54,73,86,57,70,89)(59,79,91,63,75,95)(60,80,92,64,76,96)\\ \ (97,167,113,122,129,166)(98,168,114,121,130,165)(99,174,115,123,131,176)\\ \ (100,173,116,124,132,175)(101,137,157,192,106,141)(102,138,158,191,105,142)\\ \ (103,133,156,183,160,187)(104,134,155,184,159,188)(107,136,111,140,152,182)\\ \ (108,135,112,139,151,181)(109,144,154,189,149,185)(110,143,153,190,150,186)\\ \ (117,146,120,162,169,178)(118,145,119,161,170,177)(125,164,172,180,127,148)\\ \ (126,163,171,179,128,147),\\ y:=(1,175,41,97,32,110,2,176,42,98,31,109)(3,118,44,148,22,152,4,117,43,\\ \ 147,21,151)(5,112,35,170,28,180,6,111,36,169,27,179)(7,134,40,187,23,184,8,\\ \ 133,39,188,24,183)(9,113,48,153,18,174,10,114,47,154,17,173)(11,163,37,108,\\ \ 19,119,12,164,38,107,20,120)(13,192,46,138,29,141,14,191,45,137,30,142)\\ \ (15,149,33,124,25,129,16,150,34,123,26,130)(49,127,89,145,80,158,50,128,90,\\ \ 146,79,157)(51,166,92,100,70,104,52,165,91,99,69,103)(53,160,83,122,76,132,\\ \ 54,159,84,121,75,131)(55,182,88,139,71,136,56,181,87,140,72,135)(57,161,96,\\ \ 105,66,126,58,162,95,106,65,125)(59,115,85,156,67,167,60,116,86,155,68,168)\\ \ (61,144,94,186,77,189,62,143,93,185,78,190)\\ \ (63,101,81,172,73,177,64,102,82,171,74,178),\\ z:=(1,6,10,3,16,11)(2,5,9,4,15,12)(7,14)(8,13)(17,22,26,19,32,27)\\ \ (18,21,25,20,31,28)(23,30)(24,29)(33,38,42,35,48,43)(34,37,41,36,47,44)\\ \ (39,46)(40,45)(49,54,58,51,64,59)(50,53,57,52,63,60)(55,62)(56,61)\\ \ (65,70,74,67,80,75)(66,69,73,68,79,76)(71,78)(72,77)(81,86,90,83,96,91)\\ \ (82,85,89,84,95,92)(87,94)(88,93)(97,131,113,99,129,115)(98,132,114,100,\\ \ 130,116)(101,170,136,111,171,141)(102,169,135,112,172,142)(103,175,186,110,\\ \ 165,187)(104,176,185,109,166,188)(105,120,181,108,125,191)(106,119,182,107,\\ \ 126,192)(117,139,151,127,138,158)(118,140,152,128,137,157)(121,183,160,124,\\ \ 190,150)(122,184,159,123,189,149)(133,156,173,143,153,168)(134,155,174,144,\\ \ 154,167)(145,179,161,147,177,163)(146,180,162,148,178,164).$

Furthermore let

$a:=(2,3,90,49,10,57,52,76,75,78,86,70,79,15,53,95,93,47,23,64,88,17,56,\\ \ 77,38,96,54,36,45,21,37,62,22,85,34,82,14,48,66,51,73,12,27,40,5,69,31,19,\\ \ 29,32,67,87,72,20,28,92,94,13,83,74,7,18,59,41,61,8,39,46,81,9,60,44,24,16,\\ \ 30,80,63,65,35,89,33)(4,91,42,58,50,11,26,84,71,68,43,25)(6,55),\\ b:=Id,\\ c:=(2,61,12,24,31,66,79,38,11,90)(3,84,22,15,82,28,21,35,52,74,18,5)\\ \ (4,60,8,29,71,81,85,95,92,59,51,93,16,57,9,27,94)(6,56,13,26,86,25,78,20,7,\\ \ 91,67,48,54,72,53,42,44,73,62,36,64,75,83,89)(10,40,76,63,88,58,34)(14,45,\\ 70,69,46,41,80,30,96,32)(17,33)(19,68,23,87,65,39,37)(43,47,50,77,55,49),\\ d:=(1,3,92,32,8,60,75,81,95,68,29,77,61,10,4,58,41,69,39,47,48,63,52,76,\\ \ 30,34,70,62,44,54,9,25,72,23,94,86,6,27,87,67,82,16,59,49,42,20,13,28,14,\\ \ 2,26,36,53,43,37,17,35,38,80,24,7,90,96,21,33,19,88,22,78,40,64,73,56,74,\\ \ 45,79,84,31,93,5)(11,91,65,46,18,71,83,66,55,51,50,12,57)(15,89).$

The orbits $a*U$, $b*U$, $c*U$, $d*U$ have cardinality $576$, $96$, $48$, $48$ respectively and the $(96,95)$ - PA $C:=a*U\cup b*U\cup c*U\cup d*U$ is $(96,8)$ - separable. This proves

Theorem 5**.**

$N(96)\geq 8$.

Similar as in the cases above the isometry group $U$ is an extensions from $\Delta(\Phi({\cal R}({\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}\times{\mathbb{Z}}_{6})))$ of index $48$. We calculated the separable orbits stabilized by a subgroup of order $8$. Again a backtrack search gave our result.

Remark 1**.**

Applying the theorems of Wilson [14] on our results there are more than $700$ new improvements for $N(n)$ where $100\leq n\leq 10000$. Since there are so many we do not list them here.

4 Some remarks on known results

We investigated almost all cases for $10\leq n\leq 100$ and were able to construct separable PAs corresponding to MOLS for the actual known best bounds for a lot of cases. Unfortunately we found improvements just for $n=35$, $n=48$, $n=63$ and $n=96$. But for some cases there are nice descriptions for the codes which we would like to present here.

For $n\in\{20,21,56\}$ there are separable $(n,m)$-PA consisting of just one orbit with representative Identity $Id$ corresponding to $4,5$ and $7$ MOLS respectively. In the following list we give the related isometry groups $U$ by generators.

$n=20$:

$\mid U\mid=80,$

\begin{array}[]{lll}U=\langle&(1,5,9,13,17)(2,6,10,14,18)(3,7,11,15,19)(4,8,12,16,20)&\\ &(21,23,32,25,26,38,35,36,37,31)(22,27,33,28,29,30,24,39,40,34),&\\ &(1,27,20,38,2,24,19,21)(3,37,17,28,4,25,18,40)(5,34,16,36,6,29,15,32)&\\ &(7,35,13,30,8,23,14,22)(9,39,12,31,10,33,11,26)\ \ \rangle&\end{array}

$n=21$:

$\mid U\mid=105,$

\begin{array}[]{lll}U=\langle&(1,10,21,18,16,5,13,4,15,20,17,7,9,11,6,12,8,2,14,19,3)&\\ &\ (23,27,31,35,39)(24,28,32,36,40)(25,29,33,37,41)(26,30,34,38,42)\ \ \rangle&\\ \end{array}

$n=56$:

$\mid U\mid=9408,$

\begin{array}[]{lll}U=\langle&(1,39,52,8,34,53)(2,37,49,7,36,56)(3,38,51,6,35,54)(4,40,50,5,33,55)&\\ &(9,15,12,16,10,13)(11,14)(17,47,28,24,42,29)(18,45,25,23,44,32)(19,46,27,22,43,30)&\\ &(20,48,26,21,41,31)(57,58,60)(61,64,63)(65,90,76)(66,92,73)(67,91,75)(68,89,74)&\\ &(69,96,79)(70,94,78)(71,93,80)(72,95,77)(81,98,108)(82,100,105)(83,99,107)&\\ &(84,97,106)(85,104,111)(86,102,110)(87,101,112)(88,103,109),&\\ &(1,27,6,26,8,31,4,25,3,30,2,32,7,28)(5,29)(9,19,14,18,16,23,12,17,11,22,10,24,15,20)&\\ &(13,21)(33,51,38,50,40,55,36,49,35,54,34,56,39,52)(37,53)(41,43,46,42,48,47,44)&\\ &(57,71,81,75,97,92,111,63,105,95,101,79,86,65)&\\ &(58,72,82,76,98,91,112,64,106,96,102,80,85,66)&\\ &(59,69,83,73,99,90,109,61,107,93,103,77,88,67)&\\ &(60,70,84,74,100,89,110,62,108,94,104,78,87,68)\ \ \rangle&\end{array}

For $n=20$ our PA consists of just one orbit of size $80$ and the PA is extendable to a $(20,19)$ - PA of size $96$. The extra part is a $(20,20)$-PA. The four MOLS do not belong to a difference matrix.

In the case $n=21$ the PA can be regarded as the double coset $HIdK$ where $H$ is isomorphic to $\mathbb{Z}_{21}$ and $K$ is isomorphic to $\mathbb{Z}_{5}$. It turned out that this is a conversion from a result of A.V. Nazarok [11].

For $n=14$ we found two different isometry groups, one of order $14$ and one of order $21$. It turned out that the PA for the group of order $21$ corresponds to the difference matrix given by D.T. Todorov [13]. The orbit sizes are $21,21,7$ and $7$. In the following this group is given by generators and the orbits are given by representatives in the set $R$.

$\mid U\mid=21,$

\begin{array}[]{lll}U=\langle&(2,3,5)(4,7,6)(9,10,12)(11,14,13)(16,17,19)(18,21,20)(23,24,26)(25,28,27),&\\ &(1,2,3,4,5,6,7)(8,9,10,11,12,13,14)(15,16,17,18,19,20,21)(22,23,24,25,26,27,28)&\rangle,\\ R=\{&(1,2,4,3,12)(5,10,14,8,7)(6,9)(11,13),(1,10,8)(2,9,6,14)(4,5)(7,12,11),&\\ &(1,6,14,8,3,2,13,10,4,7,9,11)(5,12),(1,2,8,14,5,7,12,10,6,3,13,9)&\}\\ \end{array}

5 Acknowledgements

We told Julian Abel about our improvements and like to thank him for the kind discussions via Email.