On Small Automorphism Groups of Binary Self-Dual Codes

1. Introduction

Binary self-dual codes are important objects in Coding Theory due to their good error-correcting properties. One possibility to classify and to characterize them is to investigate their automorphism groups. Automorphism groups of binary self-dual codes were studied for example in [1,2,3]. Those binary self-dual codes which have the highest possible minimum distance appear to have very large automorphism groups, like the famous Golay code whose automorphism group is the Mathieu group ${\mathcal{M}}_{24}$ [4]. In this paper, we investigate how small automorphism groups of binary self-dual codes can be generated. The existence of a binary self-dual code of length 72 with highest possible minimum distance has been an open question [5] for a long time. For its solution several monetary prizes are proposed [6]. The paper is structured as follows. In Section 2 we give all necessary definitions and recall classical coding theoretical results that are important for our proofs. In Section 3 we show on which sets of coordinates can act an automorphism. In Section 4 we investigate binary self-dual codes whose automorphism group can be generated by exactly one transposition or by exactly one 3-cycle permutation. Finally, in Section 6 we give new restrictions for the construction of a self-dual binary $(72,36,16)$-code based on its mutual automorphism group.

2. Preliminaries

We denote the finite field of two elements by ${\mathbb{F}}_2$ and the n-dimensional vectorspace over ${\mathbb{F}}_2$ by $V^n\left({\mathbb{F}}_2\right).$ A binary (linear) code of dimension k is a k-dimensional subspace of $V^n\left({\mathbb{F}}_2\right).$ The elements of a code are called codewords. The weight of a codeword is the number of its nonzero coordinates. The weight of a linear code is the minimum of all of the weights of all of its codewords. The weight of a code is also called minimum distance and is denoted by $d.$ Usually, a linear code is denoted as $(n,k,d)$-code, where n is the codelength, k the dimension as subspace and d the minimum distance. The inner product of two codewords is computed as the sum of all coordinatewise products, i.e. if C is a linear code and $c_1,c_2\in C,$ then the inner product of $c_1$ and $c_2$ is computed by

$$\langle c_1,c_2\rangle =\sum _{i=1}^n{c}_1\left[i\right]·c_2\left[i\right].$$

We say that $c_1$ and $c_2$ are orthogonal to each other if $\langle c_1,c_2\rangle =0.$ The dual code of C is denoted by $C^{\perp }$ and is defined by

$$C^{\perp }=\{u\mid u\in V^n\left({\mathbb{F}}_2\right) \mathrm{and} \langle u,c\rangle =0forallc\in C\}.$$

A code is called self-dual if $C=C^{\perp }.$ For two codewords $c_1$ and $c_2,$ we denote the number of coinciding 1 coordinates by $\mu (c_1,c_2)=\#\{i\mid c_1\left[i\right]=c_2\left[i\right]=1\}.$ If C is self-dual, then for every $c_1,c_2\in C$ we have $\mu (c_1,c_2)\equiv 0mod2.$

Every $k\times n$ matrix that is a basis for C as subspace is called generator matrix of $C.$

Let $\sigma$ be a permutation on a set of n elements. We apply permutations on the columns of matrices. If C is a linear code with generator matrix $G,$ then $G^{\sigma }$ generates a code $C_{\sigma }$ which is permutation equivalent to $C.$ If $C_{\sigma }=C,$ then $\sigma$ is an automorphism of $C.$ All automorphisms of a linear code form a group, called the automorphism group of C denoted by $Aut\left(C\right).$

It is well known that every linear code C is permutation equivalent to a code generated by a matrix in standard form $G_C=(I_k,A),$ where $I_k$ denotes the identity matrix of order k and A is a $k\times (n-k)$ matrix. If C is generated by a matrix in standard form, then its dual $C^{\perp }$ is generated by a matrix of the form $H_C=(-A^t,I_{n-k}),$ where $A^t$ denotes the transponed of $A.$ Note, that if C is binary, then $-A=A.$

3. Automorphisms Acting on the Coordinates

We assume that C is a self-dual binary code generated by a matrix in standard form. One may wonder then if an automorphism of C may act only on one half of the coordinates, namely on the part of the identity matrix, either in $G_C=(I_k,A)$ or in $H_C=(-A^t,I_{n-k}).$ In the following we show that this is not possible and every automorphism of C is acting on both halfs.

Theorem 1. 

Let C be a self-dual binary $(n,k,d)$-code generated by a matrix in standard form with $d>2.$ Then every non trivial automorphism of C acts on $\{1,\dots ,k\}$ and on $\{k+1,\dots ,2k=n\}.$

Proof. 

We assume indirectly that there exists $\sigma \in Aut\left(C\right)$ acting only on $\{1,\dots ,k\}.$ By assumption, C can be generated by $G_C=(I_k,A),$ and we denote

$$I=\left(\begin{array}{c}{i}_1\\ ⋮\\ i_k\end{array}\right),G=\left(\begin{array}{c}{g}_1\\ ⋮\\ g_k\end{array}\right) \mathrm{and} A=\left(\begin{array}{c}{a}_1\\ ⋮\\ a_k\end{array}\right).$$

Then $G_C^{\sigma }=(I_k^{\sigma },A).$ Since $\{g_1,\dots ,g_k\}$ are codewords of C and $\sigma$ acts on at least two coordinates - denotes by s and t -, then $g_s=(i_s,a_s),$$g_t=(i_t,a_t)$ and we have $g_s^{\sigma }=(i_t,a_s).$ But there is only one codeword with exactly one 1 in the first k coordinates in coordinate $t.$ Therefore, $a_t=a_s.$ This implies $d=2,$ which is a contradiction. If $\sigma$ is acting on $\{k+1,\dots ,2k=n\},$ then we have a similar discussion since $H_c=(A^t,I_k)$ generates C as well. □

In order to standardize a generator matrix of a code we can use the well-known Gauß elimination, which allows to add rows to each other and the change of rows (note that column change is not allowed).

Theorem 2. 

Let C be a binary self-dual $(n,k,d)$-code with $d\ge 4$ and generator matrix $G_C=(I_k,A).$ Then $\sigma \in Aut\left(C\right)\iff G_C=Gauß\left(G_c^{\sigma }\right).$

Proof. 

First, we prove the direction ⇒: Applying $\sigma$ to the columns of $G_C,$ we get a matrix in non-standard form. (By Theorem 1 we know that $\sigma$ cannot act only on $A.$ ) We apply now the steps of Gauß elimination in order to achieve $I_k$ on the lefthand side. Since $\sigma \in Aut\left(C\right),$ the set of codewords generated by $G_C$ and $G_C^{\sigma }$ coincide. Therefore, every codeword with only one 1 in the first k coordinates is uniquely determined. Thus $Gauß\left(G_C^{\sigma }\right)=G_C.$ The other direction ⇐ follows from the fact that $Gauß\left(G_C^{\sigma }\right)$ and $G_C$ generate the same code, thus by definition $\sigma \in Aut\left(C\right).$ □

4. Small Automorphism Groups

4.1. Automorphism Groups Generated by One Transposition

Let C be a linear code. If $Aut\left(C\right)=C_2=\langle \left(ab\right)\rangle ,$ then there is exactly one transposition (i.e. switch of columns) which fixes the set of codewords of $C.$ In the following we show that this case is impossible if C is a binary self-dual code.

Theorem 3. 

Let C be a self-dual binary code of length n and $\left(ab\right)$ a transposition acting on the coordinates of $C.$ Then $Aut\left(C\right)\ne \langle \left(ab\right)\rangle .$

Proof. 

We assume indirectly, that there exists $C-(n,k,d)$ self-dual binary code with generator matrix $G_C=(I_k,A)$ and automorphism group isomorphic to $\langle \sigma =\left(ab\right)\rangle .$ Without loss of generality, we may assume by Theorem 1 that $a\in \{1,\dots ,k\}$ and $b\in \{k+1,\dots ,2k=n\}.$ With the notations

$$I=\left(\begin{array}{c}{i}_1\\ ⋮\\ i_k\end{array}\right),G=\left(\begin{array}{c}{g}_1\\ ⋮\\ g_k\end{array}\right) \mathrm{and} A=\left(\begin{array}{c}{a}_1\\ ⋮\\ a_k\end{array}\right),$$

we then can differ into two cases.

First case, we assume $g_a\left[b\right]=0.$ We know that $g_a\left[a\right]=1$ and that there exists $t\ne a,$ such that $g_t\left[b\right]=1.$ But $g_t\left[a\right]=0,$ since $i_t\left[a\right]=0$ iff $t\ne a.$ Then $g_t^{\sigma }\left[a\right]=g_t\left[b\right]=1$ and $g_a^{\sigma }\left[a\right]=g_a\left[b\right]=0.$ Now, we consider the codeword with $g_t\left[t\right]=1$ and

$$g_t\left[a\right]=1 \mathrm{and} g_t\left[j\right]=0 \mathrm{for} j\in \{1,\dots ,k\}\setminus \{t,a\}$$

(1)

There is exactly one codeword c with this property. We have $c=g_t+g_a.$ Also, $g_t^{\sigma }$ fulfills property 1, since $i_t^{\sigma }\left[t\right]=1,i_t^{\sigma }\left[a\right]=1$ and $i_t^{\sigma }\left[j\right]=0\forall j\in \{1,\dots ,k\}\setminus \{t,a\}.$ But, $g_t^{\sigma }\left[b\right]=0,$ whereas $g_t\left[b\right]+g_a\left[b\right]=1.$ Therefore, $c\ne g_t^{\sigma },$ and we reached a contradiction.

Second case, we assume $g_a\left[b\right]=1.$ Since there are at least $d-1$ 1’s in column $b,$ there exists t such that $g_t\left[b\right]=1.$ Again, we have $g_a\left[a\right]=1.$ Therefore, the codeword $c=g_t+g_a$ is the only codeword with $c\left[a\right]=c\left[t\right]=1$ and $c\left[j\right]=0\forall j\in \{1,\dots ,k\}\setminus \{t,a\}.$ Since $\mu (g_i,g_j)\equiv 0mod2,$ for all possible indices $i,j$ we have that $\mu (a_a,a_t)\ge 2.$ Thus, there exists $s\in \{k+1,\dots ,2k\}\setminus \left\{b\right\}$ such that $g_a\left[s\right]=g_t\left[s\right]=1$ and thus $c\left[s\right]=0.$ But $g_t^{\sigma }\left[a\right]=g_t\left[b\right]=1,$ and therefore $g_t^{\sigma }$ is the codeword fulfilling $g_t^{\sigma }\left[t\right]=g_t^{\sigma }\left[a\right]=1$ and $g_t^{\sigma }\left[j\right]=0\forall j\in \{1,\dots ,k\}\setminus \{t,a\}.$ Further, $g_t^{\sigma }\left[s\right]=g_t\left[s\right]=1,$ thus $g_t^{\sigma }\ne c,$ therefore we get a contradiction. □

4.2. Automorphism Groups Generated by Exactly One 3-Cycle

Theorem 4. 

Let C be a self-dual binary code of length n and $\left(abc\right)$ a transposition acting on the coordinates of $C.$ Then $Aut\left(C\right)\ne \langle \left(abc\right)\rangle .$

Proof. 

We may assume that C can be generated by $G_C=(I_k,A)$ and $H_C=(A^T,I_k).$ Then, either in $G_C$ or in $H_C,$ the 3-cycle $\left(abc\right)$ is acting on only one column of the identity matrix. Therefore, similarly to the proof of Theorem 3, we get a contradiction. □

5. Self-Dual Codes with Trivial Automorphism Group

In this section we assume, that C is a binary self-dual code with $Aut\left(C\right)=\langle 1\rangle .$ This means that for every possible permutation $\sigma$ on a set of n elements, applying $\sigma$ to the codewords of $C,$ then there is at least one $c\in C$ such that $c^{\sigma }\notin C.$

Theorem 5. 

Let C be a binary self-dual $(n,k,d)$-code generated by $G_C=(I_k,A).$ Further, we assume $Aut\left(C\right)=\langle 1\rangle .$ Then $\{u\mid uisarowofA\}\ne \{v\mid visarowof{A}^t\}.$

Proof. 

We know that $G_C$ and $H_C$ both generate $C.$ Since $Aut\left(C\right)$ is trivial, there does not exist a permutation $\sigma \in S_n$ such that $H_C=G_C^{\sigma }.$ Since $(1,k+1)(2,k+2)\dots (k,n)$ is a permutation mapping $I_k$ in $G_C$ to $I_k$ in $H_C,$ there must not exist a permutation mapping the rows of A to the rows of $A^t.$ □

Corollary 1. 

If C is a binary self-dual $(n,k,d)$-code and $Aut\left(C\right)=\langle 1\rangle .$ Let $G_C=\left(\begin{array}{cc}{I}_k& A\end{array}\right)$ be a generator matrix of C. Then A is not symmetric.

6. Binary Self-Dual Doubly-Even Code of Length 72

The question if a self-dual binary doubly-even $(72,36,16)$-code exists has its origin in [7]. Doubly even means that all weights of all codewords are divisible by $4.$ Until today, the highest minimum distance for known binary self-dual codes of length 72 is $12.$ And the existence of such a code with minimum distance 16 is still unsolved. Over the years, this problem has got some attention among coding theorists, and even monetary prizes were called out for its solution [6]. The automorphisms of a possibly optimal binary self-dual code of length 72 were studied in [8,9,10,11,12]. As the strongest result in this series of studies, the structure of the automorphism group of an optimal binary self-dual code of length 72 can be determined to be one of the following few possibilities.

Theorem 6 

([12]). Let C be a self-dual $(72,36,16)$ code. Then $Aut\left(C\right)$ is trivial or isomorphic to $C_2,C_3,C_2\times C_2$ or $C_5.$

In the following we try to put some light on the case if $C-(72,36,16)$ exists with $Aut\left(C\right)$ isomorphic to the cyclic group of order $5.$

Theorem 7. 

Let C be a binary self-dual $(72,36,16)$-code generated by $G_C=(I_{36},A)$ with $Aut\left(C\right)=C_5$ and exactly two fixpoints u and v with $u,v\in \{1,\dots ,k\}$ or $u,v\in \{k+1,\dots ,2k=n\}.$ Then $G_C$ has two rows of weight 16 and no common 1’s, i.e. $\mu (g_u,g_v)=0.$

Proof. 

It is clear that if $Aut\left(C\right)=C_5,$ then at least two coordinates are fix (i.e. not moved by any automorphism), since $Aut\left(C\right)$ is generated by a permutation of order $5,$ i.e. all powers of this permutation act on 5-sets. Further, we assume that there are exactly two fixpoints, u and v and we assume first that $u,v\in \{1,\dots ,k\}.$ With the notations

$$I=\left(\begin{array}{c}{i}_1\\ ⋮\\ i_{36}\end{array}\right),G=\left(\begin{array}{c}{g}_1\\ ⋮\\ g_{36}\end{array}\right) \mathrm{and} A=\left(\begin{array}{c}{a}_1\\ ⋮\\ a_{36}\end{array}\right),$$

and $Aut\left(C\right)=\langle \sigma \rangle$ we have $i_u^{{\sigma }^j}=i_u$ and $i_v^{{\sigma }^j}=i_v$ for any power j of $\sigma .$ Since $g_u$ and $g_v$ are the only codewords with exactly one 1 in the first k coordinates, namely in coordinate u (respectively v), we get that $a_u^{{\sigma }^j}=a_u$ and $a_v^{{\sigma }^j}=a_v.$ Thus the 5-sets on which the powers of $\sigma$ act under $A,$ are all-1 or all-0 in $a_u$ and $a_v.$ We know that $\mu (a_u,a_v)\equiv 0mod2$ and $w(a_u+a_v)\ge 14\equiv 2mod4.$ Thus, if there are no more fixpoints, then the number of 1’s in $a_u$ and $a_v$ are congruent to 0 modulo $5.$ We need $w\left(a_i\right)+1\equiv 0mod4\Rightarrow w\left(a_i\right)\equiv 3mod4.$ Thus $w\left(a_u\right),w\left(a_v\right)\in \{35,15\}.$ If $w\left(a_u\right)=35,$ then in row $v,$ the 5-sets on which $\sigma$ is acting consist of either all-1 or all-0. But $\mu (a_u,a_v)\equiv 0mod2.$ Therefore, the only possibility is $w\left(a_v\right)=21,$ but then $w\left(g_v\right)=22,$ which is a contradiction to the fact, that C is a doubly-even code. Thus $w\left(a_u\right)=w\left(a_v\right)=15$ and $\mu (a_u,a_v)=0.$ is the only possibility. For $u,v\in \{k+1,\dots ,2k=n\}$ the discussion is similar since C can be also generated by $H_C=(A^t,I_{36}).$ □

7. Discussion

In this paper, small automorphism groups of binary self-dual codes are investigated and new results are presented. We have proved that the automorphism group cannot be generated by only one cycle permutation, neither of length 2, nor of length $3.$ No automorphism can act only on one half of the coordinates. Further, some restrictions for binary self-dual codes with trivial automorphism groups are given. These new findings are especially important for the construction of such codes. This question is extremely interesting in cases where the existence of a mutual binary self-dual code is still an open question, like for $n=72,k=36,d=16.$ Our current findings on such a code with automorphism group isomorphic to the cyclic group of order $5,$ can help in the construction of such a code and for exhaustive computational research.