Perfect codes from PGL(2,5) in Star graphs

TL;DR

This paper constructs new perfect codes in Star graphs using the embedding of PGL(2,5) into Sym_6, expanding the known classes of perfect codes in these graphs.

Contribution

It introduces a novel class of perfect codes in Star graphs derived from the cosets of PGL(2,5) embedded in Sym_6, for all n ≥ 6.

Findings

Existence of perfect codes from PGL(2,5) in S_6 and higher

Construction of perfect codes as unions of cosets

Extension of known perfect code classes in Star graphs

Abstract

The Star graph $S_n$ is the Cayley graph of the symmetric group $Sym_n$ with the generating set $\{(1\mbox{ }i): 2\leq i\leq n \}$. Arumugam and Kala proved that $\{\pi\in Sym_n: \pi(1)=1\}$ is a perfect code in $S_n$ for any $n, n\geq 3$. In this note we show that for any $n, n\geq 6$ the Star graph $S_n$ contains a perfect code which is a union of cosets of the embedding of $PGL(2,5)$ into $Sym_6$.

Equations

$$C^n=C\cup \bigcup _{2\le i\le n-1}(i\text{mbox}n)C$$

Full text

Perfect codes from PGL(2,5) in Star graphs

I.Yu.Mogilnykh

Ivan Yurevich Mogilnykh

iii Novosibirsk State University,

iii Pirogova st., 1,

iii 630090, Novosibirsk, Russia

iii Sobolev Institute of Mathematics,

iii pr. Koptyuga, 4,

iii 630090, Novosibirsk, Russia

ivmog84@gmail.com

Mogilnykh, I. Yu., Perfect codes from PGL(2,5) in Star graphs

© 2019 Mogilnykh I.Yu.

This work was partially supported by the Grant RFBR 18-01-00353A

Received January, 1, 2015, published March, 1, 2015.

Abstract. The Star graph $S_{n}$ is the Cayley graph of the symmetric group $\mathrm{Sym}_{n}$ with the generating set $\{(1\mbox{ }i):2\leq i\leq n\}$. Arumugam and Kala proved that $\{\pi\in\mathrm{Sym}_{n}:\pi(1)=1\}$ is a perfect code in $S_{n}$ for any $n,n\geq 3$. In this note we show that for any $n,n\geq 6$ the Star graph $S_{n}$ contains a perfect code which is a union of cosets of the embedding of $\mathrm{PGL}(2,5)$ into $\mathrm{Sym}_{6}$.

Keywords: perfect code, efficient dominating set, Cayley graph, Star graph, projective linear group, symmetric group.

1. Introduction

Let $G$ be a group with an inverse-closed generating set $H$ that does not contain the identity. The Cayley graph $\Gamma(G,H)$ is the graph whose vertices are the elements of $G$ and the edge set is $\{(hg,g):g\in G,h\in H\}$. The symmetric group of degree $n$ is denoted by $\mathrm{Sym}_{n}$. The stabilizer of an element $i\in\{1,\ldots,n\}$ by $\mathrm{Sym}_{n}$ is denoted by $Stab_{i}(\mathrm{Sym}_{n})$. The Star graph $S_{n}$ is $\Gamma(\mathrm{Sym}_{n},\{(1\mbox{ }i):2\leq i\leq n\})$.

A code in a graph $G$ is a subset of its vertices. The size of $C$ is $|C|$. The minimum distance of a code is $d=min_{x,y\in C,x\neq y}d(x,y)$, where $d(x,y)$ is the length of a shortest path connecting $x$ and $y$. A code $C$ is perfect (also known as efficient dominating set) in a $k$-regular graph $\Gamma$ with vertex set $V$ if it has minimum distance 3 and the size of $C$ attains the Hamming upper bound, i.e. $|C|=|V|/(k+1)$. We say that two codes in a graph $\Gamma$ are isomorphic if there is an automorphism of the graph $\Gamma$ that maps one code into another.

Let $T_{0}$, $T_{1}$ be distinct subsets of vertices of a graph $\Gamma$. The ordered pair $(T_{0},T_{1})$ is called a perfect bitrade, if for any vertex $x$, the set consisting of $x$ and its neighbors in $\Gamma$ meets $T_{0}$ and $T_{1}$ in the same number of vertices that is zero or one. The size of $|T_{0}|$ is called the volume of the bitrade. In particular, if $C$ and $C^{\prime}$ are perfect codes in $\Gamma$, then $(C\setminus C^{\prime},C^{\prime}\setminus C)$ is a perfect bitrade. In this case the bitrade $(C\setminus C^{\prime},C^{\prime}\setminus C)$ is called embeddable into a perfect code. In general, bitrades (non necessarily perfect) are often associated with classical combinatorial objects such as perfect codes, Steiner triple and quadruple systems and latin squares (e.g. see a survey [10]). Bitrades are used in constuctions of the parent combinatorial objects or for obtaining upper bounds on their number.

The first well-known error-correcting code was the binary Hamming code. This code is a perfect code in the Hamming graph, which is a Cayley graph of the group $Z_{2}^{n}$. Later in [14] Vasiliev showed that there are perfect codes that are nonisomorphic to the Hamming codes. A somewhat similar fact holds for the Star graph as in Section 3 we show that there are perfect codes nonisomorphic to the first series of perfect codes in the Star graph from [3].

Generally speaking, the permutation codes are subsets of $\mathrm{Sym}_{n}$ with respect to a certain metric. These codes are of practical interest for their various applications in areas such as flash memory storage [13] and interconnection networks [1]. The permutation codes with the Kendall $\tau$-metric (i.e. codes in the bubble-sort graph $\Gamma(\mathrm{Sym}_{n},\{(i\mbox{ }i+1):1\leq i\leq n-1\})$) were considered by Etzion and Buzaglo in [11]. They showed that no perfect codes in these graphs exist when $n$ is prime or $4\leq n\leq 10$. In [12] the nonexistence of the perfect codes in the Cayley graphs $\Gamma(\mathrm{Sym}_{n},H)$ was established, where $H$ are transpositions that form a tree of diameter 3.

The spectral graph theory is important from the point of view of coding theory. In particular, according to the famous Lloyd’s theorem the existence of a perfect code in a regular graph necessarily implies that $-1$ is an eigenvalue of the graph. The integrity of the spectra of several classes of Cayley graphs of the symmetric and the alternating groups was proven in [7]. The eigenvalues of $S_{n}$ are all integers $i,-(n-1)\leq i\leq(n-1)$ that follows from the spectra of the Jucys-Murphy elements [8]. The multiplicities of the eigenvalues of $S_{n}$ were studied in [2] and the second largest eigenvalue $n-2$ was shown to have multiplicity $(n-1)(n-2)$. In [5] an explicit basis for the eigenspace with eigenvalue $n-2$ was found and a reconstruction property for eigenvectors by its partial values was proven. Later in [6] it is shown that the basis consists of eigenvectors with minimum support.

For $l,r\in\mathrm{Sym}_{n}$ define the following mapping on the vertices of $S_{n}$: $\lambda_{l,r}(g)=lgr$, $g$ in $\mathrm{Sym}_{n}$.

Theorem 1**.**

[9]** The automorphism group of $S_{n}$ is $\{\lambda_{l,r}:l\in Stab_{1}(\mathrm{Sym}_{n}),r\in\mathrm{Sym}_{n}\}$.

In [3] Arumugam and Kala showed that $Stab_{1}(\mathrm{Sym}_{n})$ is a perfect code in $S_{n}$, for any $n\geq 3$ . Consider the isomorphism class of $Stab_{1}(\mathrm{Sym}_{n})$ in $S_{n}$. By Theorem 1 the only left multiplication automorphisms are those by the elements from $Stab_{1}(\mathrm{Sym}_{n})$. Therefore we have the following result.

Corollary 1**.**

The isomorphism class of $Stab_{1}(\mathrm{Sym}_{n})$in $S_{n}$ is the set of its right cosets in $\mathrm{Sym}_{n}$.

In Section 2 we prove that the projective linear group $\mathrm{PGL}(2,5)$ is a perfect code, which is isomorphic to $\{\pi\in\mathrm{Sym}_{6}:\pi(1)=1\}$ as a group via an outer automorphism of $\mathrm{Sym}_{6}$, but is nonisomorphic to it with respect to the automorphism group of the Star graph. We continue the study in Section 3 where we construct a new series of perfect codes in Star graphs $S_{n}$, $n\geq 7$ using cosets of $\mathrm{PGL}(2,5)$. Also we obtain the classification of the isomorphism classes of perfect codes and perfect bitrades in Star graphs $S_{n}$, $n\leq 6$ by linear programming.

2. Perfect codes from $\mathrm{PGL}(2,5)$ in $S_{6}$

The action of a group $G$ on a set $M$ is regular if it is transitive and $|G|=|M|$, i.e. for any $x$, $y\in M$ there is exactly one element of $G$ sending $x$ to $y$.

Let $\mathrm{PGL}(n,q)$ be the projective linear group induced by the action of $\mathrm{GL}(n,q)$ on the 1-dimensional subspaces (projective points) of a $n$-dimensional space over the field of order $q$. It is well known that $\mathrm{PGL}(n,q)$ acts transitively on the ordered pairs of distinct projective points for $n\geq 3$ and regularly on the ordered triples of pairwise distinct projective points when $n=2$, see e.g. [4][Exercises 2.8.2 and 2.8.7].

Proposition 1**.**

The group $\mathrm{PGL}(2,q)$ acts regularly on the ordered triples of distinct projective points.

In throughout what follows we enumerate the projective points by the elements of $\{1,\ldots,6\}$, so $\mathrm{PGL}(2,5)$ is embedded in $\mathrm{Sym}_{n}$, $n\geq 6$. An element of $\mathrm{Sym}_{n}$ is a cycle of length $m$, if it permutes $i_{1},\ldots,i_{m}\in\{1,\ldots,n\}$ in the cyclic order and fixes every element of $\{1,\ldots,n\}\setminus\{i_{1},\ldots,i_{m}\}.$

Corollary 2**.**

The group $\mathrm{PGL}(2,5)$ does not contain cycles of length $2$ or $3$.

Proof..

By Proposition 1 the group $\mathrm{PGL}(2,5)$ is regular on the triples of the elements of $\{1,\ldots,6\}$. In particular, any permutation of $\mathrm{PGL}(2,5)$ that has at least three fixed projective points is the identity. We conclude that there are no cycles of length 2 or 3 in $\mathrm{PGL}(2,5)$ since they have three fixed points. ∎

Lemma 1**.**

Let $\pi$ be a permutation from $\mathrm{Sym}_{n}$, $n\geq 6$. Then $\pi\mathrm{PGL}(2,5)$ is a code in $S_{n}$ with the minimum distance 3.

Proof..

Suppose that $\pi\pi^{\prime}$ and $\pi\pi^{\prime\prime}$ are adjacent in $S_{n}$, $\pi^{\prime}$, $\pi^{\prime\prime}\in\mathrm{PGL}(2,5)$. Then by the definition of the Star graph $S_{n}$ there is $x$, $2\leq x\leq n$ such that $(1\mbox{ }x)\pi\pi^{\prime}=\pi\pi^{\prime\prime}$, so $\pi^{-1}(1\mbox{ }x)\pi=\pi^{\prime\prime}(\pi^{\prime})^{-1}$ is in $\mathrm{PGL}(2,5)$. This contradicts Corollary 2 because $\pi^{-1}(1x)\pi$ is a transposition. If $\pi\pi^{\prime}$ and $\pi\pi^{\prime\prime}$ are at distance 2 in $S_{n}$, then there are $x$ and $y$, $2\leq x,y\leq n$, $x\neq y$ such that $\pi^{-1}(1\mbox{ }x)(1\mbox{ }y)\pi$ is in $\mathrm{PGL}(2,5)$. So, $\pi^{-1}(1\mbox{ }x)(1\mbox{ }y)\pi$ is a cycle of length 3, which contradicts Corollary 2.

∎

Theorem 2**.**

The group $\mathrm{PGL}(2,5)$ is a perfect code in $S_{6}$ and the partitions of $\mathrm{Sym}_{6}$ into the left and into the right cosets by $\mathrm{PGL}(2,5)$ are partitions of the Star graph $S_{6}$ into perfect codes.

Proof..

The order of $\mathrm{PGL}(2,5)$ is $5!$, which is the size of a perfect code in $S_{6}$ by the Hamming bound. Lemma 1 implies that $\mathrm{PGL}(2,5)$ as well as any left coset of $\mathrm{PGL}(2,5)$ is a perfect code. Since the right multiplication by any element of $S_{n}$ is an automorphism of $S_{n}$ by Theorem 1, every right coset of $\mathrm{PGL}(2,5)$ is also a perfect code. The partitions into the left and right cosets are different because $\mathrm{PGL}(2,5)$ is not a normal subgroup in $\mathrm{Sym}_{6}$.

∎

3. Recursive construction for perfect codes in the Star graphs from $\mathrm{PGL}(2,5)$

Let $C$ be a code in $S_{n}$. For a permutation $\sigma$ from $Sym(n)$ denote by $\sigma C=\{\sigma\pi:\pi\in C\}$. If $\sigma$ fixes 1 by Theorem 1 the left multiplication by $\sigma$ is an automorphism of $S_{n}$ and therefore the set of distances between any two permutations of $C$ coincides with that of $\sigma C$. In this section we show that a code in the Star graph $S_{n-1}$ with minimum distance three could be embedded into a code in the Star graph $S_{n}$ with minimum distance three by taking $(n-1)$ left multiplications of $C$ by transpositions. In particular, we obtain a new infinite series of perfect codes in the Star graphs $S_{n}$ from $\mathrm{PGL}(2,5)$ for any $n,n\geq 6$.

Theorem 3**.**

Let $C$ be a code with minimum distance 3 in $S_{n-1}$. Then the code

[TABLE]

is a code of size $|C|(n-1)$ with minimum distance 3.

Proof..

We introduce an auxilary notation and prove a technical result. Let $\Gamma_{i}$ denote the subgraph of $S_{n}$ induced by the set of vertices $(i\mbox{ }n)\mathrm{Sym}_{n-1}$, $i\in{1,\ldots,n-1}$, $\Gamma_{n}$ denote the subgraph of $S_{n}$ induced by the vertices from $\mathrm{Sym}_{n-1}$. Note that in [5] (see also [6][Section 6]) a similar partition was considered for constructing a basis for eigenspace of $S_{n}$ corresponding to eigenvalue $n-2$.

Lemma 2**.**

1. For any $i,2\leq i\leq n$, $\Gamma_{i}$ is an isometric subgraph of $S_{n}$ that is isomorphic to $S_{n-1}$. The set of vertices of $\Gamma_{1}$ is a perfect code in $S_{n}$.

2. Let $\pi$ be a permutation from $\mathrm{Sym}_{n-1}$. Then for any i, $2\leq i\leq n-1$ the vertex $(i\mbox{ }n)\pi$ of $\Gamma_{i}$ has exactly one neighbor in $S_{n}$ outside of $\Gamma_{i}$ and it is the vertex (1 n)(1 i) of $\Gamma_{1}$. The only neighbor of $\pi$ in $S_{n}$ outside $\Gamma_{n}$ is $(1\mbox{ }n)\pi$.

Proof..

1. Obviously, the vertices of $\mathrm{Sym}_{n-1}$ induce an isometric subgraph of $S_{n}$ which is isomorphic to $S_{n-1}$. By Theorem 1 the left multiplication by $(i\mbox{ }n)$ is an automorphism of $S_{n}$ for any $i\in\{2,\ldots,n\}$. We conclude that $\Gamma_{i}$ are isomorphic copies of $S_{n-1}$ for any $i\in\{2,\ldots,n\}$. By Corollary 1 we have that $(Stab_{1}(\mathrm{Sym}_{n}))(1\mbox{ }n)=(1\mbox{ }n)\mathrm{Sym}_{n-1}$ is a perfect code in $S_{n}$. Since this set is exactly the vertices of $\Gamma_{1}$, we obtain the required.

2. Since $\Gamma_{i}$ is isomorphic to $S_{n-1}$, it is $(n-2)$-regular for $i\in\{2,\ldots,n-1\}$. The remaining neighbor of $(i\mbox{ }n)\pi$ outside $\Gamma_{i}$ is the vertex $(1\mbox{ }i)(i\mbox{ }n)\pi=(1\mbox{ }n)(1\mbox{ }i)\pi$ of $\Gamma_{1}$. ∎

Obviously, the size of $C^{n}$ is $(n-1)|C|$. We now show that the minimum distance of $C^{n}$ is three. We see that each of the graphs $\Gamma_{i}$ contains the copy $(i\mbox{ }n)C$ of the code $C$, for any $i\in\{2,\ldots,n-1\}$ and $\Gamma_{n}$ contains $C$. The distances between vertices from $(i\mbox{ }n)C$ are the same as those of $C$ in $S_{n-1}$. Therefore, it remains to show that the distances between the vertices of $(i\mbox{ }n)C$ and $(k\mbox{ }n)C$ and the distances between the vertices of $(i\mbox{ }n)C$ and $C$ are at least 3, for any distinct $i,k$ such that $2\leq i,k\leq n-1$. By the second statement of Lemma 2, these distances are at least 2.

Let $(i\mbox{ }n)\pi$ and $(k\mbox{ }n)\pi^{\prime}$ be at distance 2, $\pi,\pi^{\prime}\in C$. Then by the second statement of Lemma 2 they both have a common neighbor in $\Gamma_{1}$, which is $(1\mbox{ }n)(1\mbox{ }i)\pi=(1\mbox{ }n)(1\mbox{ }k)\pi^{\prime}$. This implies that $(1\mbox{ }i)(1\mbox{ }k)\pi^{\prime}=\pi$ for $1\leq i,k\leq n-1$, or equivalently $\pi$ and $\pi^{\prime}$ are at distance 2 in $S_{n-1}$. This contradicts the minimum distance of $C$.

Let $(i\mbox{ }n)\pi$ and $\pi^{\prime}$ be at distance 2, $\pi,\pi^{\prime}\in C$. By the second statement of Lemma 2 the only neighbor of $(i\mbox{ }\pi)$ outside of $\Gamma_{i}$ is $(1\mbox{ }n)(1\mbox{ }i)\pi$ and the only neighbor of $\pi^{\prime}$ outside $\Gamma_{n}$ is $(1\mbox{ }n)\pi^{\prime}$. So we see that $(1\mbox{ }n)(1\mbox{ }i)\pi=(1\mbox{ }n)\pi^{\prime}$, which contradicts the minimum distance of $C$.

∎

Corollary 3**.**

For any $n\geq 6$ there is a perfect code in $S_{n}$ which is not isomorphic to $Stab_{1}(Sym_{n})$.

Proof..

Consider the code $D$ which is obtained by iteratively applying construction from Theorem 3 $(n-6)$ times to the code $\mathrm{PGL}(2,5)$. By the construction, the code $\mathrm{PGL}(2,5)$ is a subcode of $D$. Proposition 1 implies that there are permutations $\pi,\pi^{\prime}$ in $\mathrm{PGL}(2,5)$ such that $\pi(1)\neq\pi^{\prime}(1)$. By Corollary 1 the isomorphism class of $Stab_{1}(Sym_{n})$ in $S_{n}$ consists of its right cosets. Since we have that $\pi(1)=\pi^{\prime}(1)$ for any $\pi$ and $\pi^{\prime}$ from a right coset of $Stab_{1}(Sym_{n})$, we conclude that $D$ is not isomorphic to $Stab_{1}(Sym_{n})$. ∎

We proceed with the following computational results for small Star graphs.

Proposition 2**.**

*1. The isomorphism class of $Stab_{1}(Sym_{n})$ is the only isomorphism class of the perfect codes in $S_{n}$ for n=3,4,5.

2. The isomorphism classes of $Stab_{1}(Sym_{6})$ and $\mathrm{PGL}(2,5)$ are the only isomorphism classes of the perfect codes in $S_{6}$.*

Proof..

For $n=3$ and $4$ the uniqueness of perfect code in $S_{n}$ could be shown by hand. In case when $n=5$ and $6$ the result was obtained by binary linear programming. Because $S_{n}$ is a transitive graph, without restriction of generality, we can consider the perfect codes containing the identity permutation. In case $n=5$ there is one solution to the binary linear programming problem, which is $Stab_{1}(Sym_{n})$.

Let $n$ be six. We consider any transposition that preserves 1, say $(2\mbox{ }3)$. By the definition of the Star graph, $(2\mbox{ }3)$ is at distance three from the identity permutation. Now we split the set of all codes as follows: the codes that contain the permutation $(2\mbox{ }3)$ and those that do not. We then solve two linear programming problems separately for these cases. There are 6 solutions (perfect codes) that does not contain $(2\mbox{ }3)$. These are $\mathrm{PGL}(2,5)$ and its five conjugations. When $(2\mbox{ }3)$ is in the code, the returns with the only solution which is $Stab_{1}(Sym_{n})$.

∎

Proposition 3**.**

All perfect bitrades in $S_{n}$ are embeddable for $3\leq n\leq 6$. For $n\in\{3,4,5\}$ their volumes are equal to $(n-1)!$. For $n=6$ the volumes of bitrades are $120$, $100$ and $96$.

Proof..

The statement is obvious for $n=3$. Using linear programming approach by PC we found that for $n=4,5,6$ all bitrades are embeddable and have the corresponding volumes. When $n$ is $6$, a perfect bitrade $(C\setminus C^{\prime},C^{\prime}\setminus C)$ has volume $120$ if $C$ and $C^{\prime}$ are disjoint perfect codes, e.g. $Stab_{1}(\mathrm{Sym}_{6})$ and $Stab_{1}(\mathrm{Sym}_{6})(1\mbox{ }6)$. By Proposition 1 the group $\mathrm{PGL}(2,5)$ acts transitively on the set $\{1,\ldots,6\}$, so there are exactly $20$ permutations from $\mathrm{PGL}(2,5)$ that fix $1$. So we see that a perfect bitrade $(C\setminus C^{\prime},C^{\prime}\setminus C)$ is of volume $100$ if $C$ is $Stab_{1}(\mathrm{Sym}_{6})$ and $C^{\prime}$ is $\mathrm{PGL}(2,5)$. Finally, $(C\setminus C^{\prime},C^{\prime}\setminus C)$ is a perfect bitrade of volume $96$ if $C$ is $\mathrm{PGL}(2,5)$ and $C^{\prime}$ is one of its nontrivial conjugations. Indeed, $\mathrm{PGL}(2,5)$ is isomorphic to $\mathrm{Sym}_{5}$ via an outer automorphism of $\mathrm{Sym}_{6}$. Therefore the intersection of $\mathrm{PGL}(2,5)$ and its conjugation is a subgroup which is isomorphic to the intersection $\mathrm{Sym}_{5}$ and some of its conjugation $Stab_{i}(\mathrm{Sym}_{6})$, $i\in\{1,\ldots,5\}$. Since the latter intersection is of order 4!=24, the proposition is true.

∎

Acknowledgements The author gradually thanks Sergey Avgustinovich for introducing the problem and stimulating discussions.