On the self-dual codes with an automorphism of order 5

Abstract

For lengths 60,  62, and 64, by applying the method for constructing self-dual codes having an automorphism of odd prime order, we classify all optimal singly even self-dual codes with an automorphism of order 5 with 12 cycles. For the binary self-dual [62, 31, 12] codes we have found five new values of the parameter in the weight enumerator thus doubling the number of know values. For length 64 we have found codes with 14 new parameter values for both known weight enumerators. By shortening all binary self-dual [60, 30, 12] codes having an automorphism of order 5 we construct many new [58, 29, 10] self-dual codes. We have found a new value of the parameter in the weight enumerator of these codes.

1 Introduction

Let ${\mathbb {F}}_q$ be the finite field of q elements, for a prime power q. A linear $[n, k]_q$ code C is a k-dimensional subspace of ${\mathbb {F}}_q^n$. The elements of C are called codewords, and the (Hamming) weight of a codeword $v\in C$ is the number of the non-zero coordinates of v. We use $\text{ wt }(v)$ to denote the weight of a codeword. The minimum weight d of C is the minimum nonzero weight of any codeword in C and the code is called an $[n, k, d]_q$ code. A matrix whose rows form a basis of C is called a generator matrix of this code.

Let $(u,v)\in {\mathbb {F}}_q$ for $u,v\in {\mathbb {F}}_q^n$ be an inner product in ${\mathbb {F}}_q^n$. The dual code of an $[n,k]_q$ code C is $C^{\perp }=\{u \in {\mathbb {F}}_q^n \mid (u,v)=0$ for all $v \in C \}$ and $C^{\perp }$ is a linear $[n,n-k]_q$ code. In the binary case the inner product is the standard one, namely, $(u,v)=\sum _{i=1}^n{u_iv_i}.$ If $C \subseteq C^{\perp }$, C is termed self-orthogonal, and if $C = C^{\perp }$, C is self-dual. A binary self-dual code is doubly-even if all codewords have weight divisible by four, and singly-even if there is at least one nonzero codeword of weight $\equiv 2\pmod 4$. Self-dual doubly-even codes exist only if n is a multiple of eight.

The weight enumerator W(y) of a code C is defined as $W(y)=\sum _{i=0}^n A_iy^i$, where $A_i$ is the number of codewords of weight i in C. We say that two binary linear codes C and $C'$ are equivalent if there is a permutation of coordinates which sends C to $C'$. The set of coordinate permutations that maps a code C to itself forms a group called the automorphism group of C (denoted by $\text{ Aut }(C)$). Let $S_n$ be the symmetric group of degree n. We say that a permutation $\sigma \in S_n$ is of type $p-(c,f)$ if it has exactly c cycles of length p and f fixed point in its decomposition.

All optimal binary self-dual codes of lengths 52–60 having an automorphism of order 7 or 13 were classified in [1].

Recently, all codes of lengths $50\le n\le 60$ having an automorphism of type 5-(10, f) for $f=0,2,4,6,8$ and 10 were classified up to equivalence in [2]. For comparison reasons, we give the information for the number of inequivalent such codes, in Table 1.

Table 1 Self-dual codes with an automorphism of order 5 with 10 cycles

From [3, Table 3] we have the following cases for the length n and the type of automorphism: $n=60+2t$, type 5-(12, 2t), $t=0,1,\dots ,5$. So we have been intrigued to investigate and classify optimal self-dual codes of lengths $60\le n\le 64$ with an automorphism of order 5 with 12 cycles. To do so we continue with some properties of the binary self-dual codes having an automorphism of prime odd order.

2 Construction method

Let C be a binary self-dual code of length n with an automorphism

$$\begin{aligned} \sigma= & {} (1,2,\ldots ,p)(p+1,p+2,\ldots ,2p)\cdots (p(c-1)+1,\nonumber \\&p(c-1)+2,\ldots ,pc), \end{aligned}$$

(1)

of type $p-(c,f)$, where $f=n-pc$. Denote the cycles of $\sigma$ by $\Omega _1, \Omega _2, \ldots , \Omega _c$, and the fixed points by $\Omega _{c+1},\ldots , \Omega _{c+f}$. Let $F_\sigma (C)=\{ v \in C\mid v \sigma =v \}$, $E_\sigma (C)=\{ v\in C\mid wt(v\vert \Omega _i)\equiv 0\pmod 2,i=1,\ldots ,c+f\}$, where $v\vert \Omega _i$ is the restriction of v on $\Omega _i$.

Theorem 1

[4] Assume C is a self-dual code having an automorphism of type $p-(c,f)$. The code C is a direct sum of the subcodes $F_\sigma (C)$ and $E_\sigma (C)$. Then $F_\sigma (C)$ and $E_\sigma (C)$ are subspaces of dimensions $\frac{c+f}{2}$ and $\frac{(p-1)c}{2}$, respectively.

From the definition of $F_\sigma (C)$ it follows that $v\in F_\sigma (C)$ iff $v\in C$ and v is constant on each cycle. Let $\pi :F_\sigma (C)\rightarrow {\mathbb {F}}_{2}^{c+f}$ be the projection map where if $v\in F_\sigma (C)$, $(v\pi )_i =v_j$ for some $j\in \Omega _i, i=1,2,\ldots ,c+f$.

Denote by $E_\sigma (C)^{*}$ the code $E_\sigma (C)$ with the last f coordinates deleted. So $E_\sigma (C)^{*}$ is a self-orthogonal binary code of length pc. For v in $E_\sigma (C)^*$ we let $v\vert \Omega _i=(v_0,v_1,\ldots ,v_{p-1})$ correspond to the polynomial $v_0+v_1 x+\cdots +v_{p-1}x^{p-1}$ from ${{{\mathcal {P}}}}$, where ${{{\mathcal {P}}}}$ is the set of even-weight polynomials in ${\mathbb {F}}_2 [x]/\langle x^p-1\rangle$. Thus we obtain the map $\varphi :E_\sigma (C)^{*}\rightarrow {{{\mathcal {P}}}}^c$.

Theorem 2

[5] A binary [n, n / 2] code C with an automorphism $\sigma$ defined in (1) is self-dual if and only if the following two conditions hold:

(i) $C_{\pi }=\pi (F_\sigma (C))$ is a binary self-dual code of length $c+f$,   (ii) for every two vectors $u, v\in C_{\varphi }=\varphi (E_{\sigma }(C)^{*})$ we have $\sum \nolimits _{i=1}^{c} u_i(x)v_i(x^{-1})=0$. If  2  is a primitive root modulo p then $C_\varphi$ is a self-dual code of length c over the field ${{{\mathcal {P}}}}\cong {\mathbb {F}}_{2^{p-1}}$ under the inner product $(u,v)=\sum \nolimits _{i=1}^{c}{u_iv_i^{2^{(p-1)/2}}}.$

To classify all codes, we need additional conditions for equivalence and we use the following theorem.

Theorem 3

[6] The following transformations preserve the decomposition and send the code C to an equivalent one: (i) a permutation of the fixed coordinates;    (ii) a permutation of the p-cycles coordinates; (iii) a substitution $x\rightarrow x^2$ in $C_{\varphi }$ and (iv) a multiplication of the j-th coordinate of $C_\varphi$ by $x^{t_j}$ where $t_j$ is an integer, $0 \le t_j \le p-1$, $j=1,2,\ldots ,c$.

3 Self-dual codes with twelve cycles of length five

Let C be an optimal binary self-dual code having an automorphism of order 5 with 12 cycles and $f=2t$, $t=0,\dots ,5$ fixed points. Since 2 is a primitive root modulo 5, according to Theorem 2, the subcode $C_{\varphi }$ is a self-dual code of length c over the field ${{{\mathcal {P}}}}$ under the inner product

$$\begin{aligned} (u, v)=\sum \limits _{i=1}^{c}{u_iv_i^{4}}. \end{aligned}$$

(2)

Furthermore ${{{\mathcal {P}}}}$ is a finite field with 16 elements, $\mathcal{P}\cong {\mathbb {F}}_{16}=\{0, e=\alpha ^0, \alpha ^k| k=1,\dots ,14\}$, where $e=x+x^2+x^3+x^4$, $\alpha =1+x$ is a primitive element of multiplicative order 15. We list the elements of ${{{\mathcal {P}}}}^*$—the multiplicative group of ${{{\mathcal {P}}}}$ in Table 2. Denoting $\delta =\alpha ^5$ the group ${{{\mathcal {P}}}}^*$ can also be described as ${{{\mathcal {P}}}}^*=\{\alpha ^{3t}\delta ^l \;\vert \; 0\le t\le 4, 0\le l\le 2\}.$

Table 2 The multiplicative group of the field $\mathcal{P}^*\cong {\mathbb {F}}_{16}^*$

Proposition 1

Let $C_\varphi$ be a self-dual code of length 12 over ${{{\mathcal {P}}}}$ under the orthogonality condition (2), such that $E_\sigma (C)$ is a code with minimum weight at least 12. Then the code $C_\varphi$ has a generator matrix

$$\begin{aligned} \left( \begin{array}{c|c} eI_6 &{} \begin{array}{cccccc} t_{11} &{} \quad t_{12} &{} \quad t_{13} &{} \quad t_{14} &{} \quad t_{15} &{} \quad t_{16} \\ t_{21} &{} \quad l_{22} &{} \quad l_{23} &{} \quad l_{24} &{} \quad l_{25} &{} \quad l_{26} \\ t_{31} &{} \quad l_{32} &{} \quad l_{33} &{} \quad l_{34} &{} \quad l_{35} &{} \quad l_{36} \\ t_{41} &{} \quad l_{42} &{} \quad l_{43} &{} \quad l_{44} &{} \quad l_{45} &{} \quad l_{46} \\ t_{51} &{} \quad l_{52} &{} \quad l_{53} &{} \quad l_{54} &{} \quad l_{55} &{} \quad l_{56} \\ t_{61} &{} \quad l_{62} &{} \quad l_{63} &{} \quad l_{64} &{} \quad l_{65} &{} \quad l_{66} \\ \end{array} \end{array}\right) , \end{aligned}$$

(3)

$t_{ij}\in \{0,e,\delta ,\delta ^2\}$, $j=1,\dots ,6,$ $l_{ij}\in \mathcal{P}.$ Furthermore $(t_{11},\dots ,t_{16})$ is one of the following seven vectors $(0, 0, e, e, \delta , \delta ^2)$, $(0, e, \delta , \delta , \delta , \delta )$, $(0, e, \delta , \delta , \delta ^2, \delta ^2)$, (0, e, e, e, e, e), $(0, e, e, e, \delta , \delta )$, $(e, e, \delta , \delta , \delta , \delta ^2)$, $(e, e, e, e, \delta , \delta ^2).$

Proof

We begin by row reducing the matrix $G_\varphi .$ Using transformation (iv) from Theorem 3 we can assume that the elements in the first row of $G_\varphi$ are from the set $\{0, e,\delta ,\delta ^2\}$. Assume we use the following partial ordering in ${{{\mathcal {P}}}}$ $0\prec e\prec \delta \prec \delta ^2.$ Further interchanging the columns of $G_\varphi$, it follows that, we can take $0\preceq t_{11}\preceq t_{12}\preceq t_{13}\preceq t_{14}\preceq t_{15}\preceq t_{16}\preceq \delta ^2$. Using (2) we can reduce the vector $v=(t_{11},\dots ,t_{16})$ to cases listed in Table 3. The transformation $\gamma :x\rightarrow x^{2}$, (iii) from Theorem 3, maps $\delta$ to $\delta ^2$ and vice versa and we have $v_4\xrightarrow {\gamma }v_3,$ $v_7\xrightarrow {\gamma }v_6,$ $v_{11}\xrightarrow {\gamma }v_{10},$ $v_{13}\xrightarrow {\gamma }v_{12}.$

Obviously, the vectors $(e,0,\dots ,0,v_1)$, $\delta (e,0,\dots ,0,v_2)$, and $\delta (e,0,\dots ,0,v_3)$ have weight 8, which concludes this proof.

Since ${{{\mathcal {P}}}}^*=\{\alpha ^{3t}\delta ^l\}$, $0\le t\le 4$, $0\le l\le 2$ every element $t_{j1}\in {{{\mathcal {P}}}}^*$, $j=2,\dots , 6$ can be transformed into e, $\delta$ or $\delta ^2$ using a multiplication of j-th row of $G_\varphi$ by $\alpha ^{-3t}$, followed by some cyclic shifts in the j-th column. $\square$

Table 3 Cases for the first row of $G_\varphi$

By using a computer for calculating the possible second row of the matrix (3) we have found 242 inequivalent codes. Of these 242 codes: 66 are obtained from $v_5$, 136 from $v_6$, 123 from $v_8$, 17 from $v_9$, 136 from $v_{10},$ 193 from $v_{11},$ and 137 from $v_{14}$ (note that we have some codes that can be obtained from different first row).

Next for each of these 242 inequivalent codes we add a third row and check the result codes for minimum weight and equivalence. Of the 690,626 constructed codes there are exactly 35,191 inequivalent codes after row 3. Then for every one of these codes we add a fourth row and again check the result codes for minimum weight and equivalence. It turns out that there are exactly 681,862 inequivalent such codes (out of a total of 9,084,240 codes).

After that we added the fifth and sixth row of the matrix and check the resulted codes for equivalence and that their minimum weight is at least 12. After checking 7,197,760 codes our exhaustive computer search shows the following result.

Proposition 2

Up to permutational equivalence there are exactly 60, 467 codes $C_\varphi$ over ${{{\mathcal {P}}}}$ such that the code $\varphi ^{-1}(C_\varphi )$ has a minimum weight 12. Six of these codes have minimum weight 16 and the rest have minimum weight 12.

The number of the different values of $|\text{ Aut }(C)|$ of the constructed codes is given in Table 4.

Table 4 The order of the automorphism groups of optimal codes over ${\mathbb {F}}_{16}$

Denote by $H_i$, $i=1,\dots , 60{,}467$, the generator matrices of the codes obtained. These matrices can be obtained from [7]. For equivalence check and also for finding the weight distribution of the codes obtained we use the program Q-extensions [8] (Table 5).

Remark 1

The calculations involving the construction of the rows of the matrices $H_i$ have been performed by both authors independently. The first author used own Delphi source code for code generation, the total CPU-time for the computation was about a week on a 3 GHz processor. The second author used GAP 4.8 [9] for the generation of the codes. This computation took about two weeks. Both authors constructed the same result with a total of 60,467 codes.

4 [60, 30, 12] binary self-dual codes with an automorphism of type 5-(12, 0)

Let C be a [60, 30, 12] binary self-dual code with an automorphism of type 5-(12, 0). There are two possible forms for the weight enumerator of a binary self-dual [60, 30, 12] code [10]:

$$\begin{aligned} W_{60,1}&= 1 + 3451y^{12} + 24{,}128y^{14} + 33{,}6081y^{16} +\cdots ,\\ W_{60,2}&= 1 + (2555 + 64\beta )y^{12} + (33{,}600 - 384\beta )y^{14} +\cdots ,\quad 0\le \beta \le 10. \end{aligned}$$

A code exists for $W_{60,1}$ [10] and for $W_{60,2}$ when $\beta = 0, 1, 2, 5, 6, 7,$ and 10 [11].

Table 5 The number of optimal codes with $A_d$ over ${\mathbb {F}}_{16}$

By Theorem 2 the code $C_\pi$ is a [12, 6] binary self-dual code. There are exactly three such codes $6i_2$, $2i_2+h_8$ and $d_{12}$ (see [12]). We have that any 2-weight vector in $C_\pi$ will lead to a 10-weight vector in $F_\sigma (C)$ therefore we look for a [12, 6, 4] and thus the only possible code is $d_{12}.$

Let $Q_1$ be the automorphism group of the code $d_{12}$ with the generator matrix $G_1=\left( I_6| \begin{array}{cc} I_4 &{}\,\, A \\ A^T &{} \,\,I_4 \\ \end{array} \right) ,$ where A in a $2\times 4$ all-ones matrix. We have

$$\begin{aligned} Q_1=\langle (1,3,8)(2,7,9), (1,11,6,4,2,9)(3,7,12,5,10,8)\rangle ,\quad |Q_1|=23{,}040. \end{aligned}$$

For a permutation $\tau \in S_{12}$ denote by $C_{1,j}^{\tau }$, $j=1,\ldots ,60{,}467$ the [62, 31] self-dual code determined by the matrix $G_1$, with columns permuted by $\tau$, as a generator for $F_\sigma (C)$ and $H_j$ as a generator matrix for $E_\sigma (C)^*$. If $\tau _1$ and $\tau _2$ belong to one and the same right coset of $Q_1$ in $S_{12}$, then the codes $C_{1,j}^{\tau _1}$ and $C_{1,j}^{\tau _2}$ are equivalent. Thus we can only use the right transversal $T_1$ of $S_{12}$ with respect to $Q_1,$ we have $|T|=20{,}790.$ After calculating all codes $C_{1,j}^{\tau }$, $j=1,\ldots ,60{,}467$ for $\tau \in T_1$ we obtain the following result.

Theorem 4

Up to equivalence, there are exactly 236 optimal binary self-dual [60, 30, 12] codes having an automorphism of type 5-(12, 0).

Remark 2

All codes that we have obtained have weight enumerator $W_{60,2}.$ The number of inequivalent codes for the pairs $(\beta ,|\text{ Aut }(C)|)$ are summarized in Table 6.

Table 6 The number of codes obtained for the pair $(\beta ,|\text{ Aut }(C)|)$

Amongst codes, constructed by us, we have found 13 codes, equivalent to the codes from [13].

5 [62, 31, 12] binary self-dual codes with automorphism of type 5-(12, 2)

For the self-dual [62, 31, 12] code there are two possibilities [10]:

$$\begin{aligned} W_{62,1}&= 1 + 2308y^{12} + 23{,}767y^{14} + 279{,}405y^{16} +\cdots ,\\ W_{62,2}&= 1 + (1860 + 32\beta )y^{12} + (28{,}055 - 160\beta )y^{14} +\cdots , \end{aligned}$$

where $\beta$ is an integer parameter $0\le \beta \le 93$. Only codes with weight enumerator $W_{62,2}$ where $\beta =0,$ 9, 10, 15, 16 are known (see [3, 13, 14] and [15]).

According to Theorem 2$C_\pi$ is a [14, 7] binary self-dual code. Using [12], there are exactly four such codes, namely $7i_2,$ $3i_2\oplus e_8,$ $i_2\oplus d_{12},$ and $2e_7.$ If a 2-weight codeword occur in $C_\pi$ then the minimum weight of C is $d\le 10$ therefore only a [14, 7, 4] code can generate $C_\pi .$ Thus we have $C_\pi \cong 2e_7.$ Choosing all $\left( {\begin{array}{c}14\\ 2\end{array}}\right)$ splittings of $\{1,\dots ,14\}$ into sets $X_c$ of cyclic and $X_f$ – fixed points we found two different codes $C_\pi$ generated by $G_2=(I_7|Z_2)$ and $G_3=(I_7|Z_3),$ where $Z_2=\left( \begin{array}{ccccccc} 1&{}\,\,1&{}\,\,1&{}\,\,0&{}\,\,0&{}\,\,0&{}\,\,0\\ 0&{}\,\,1&{}\,\,1&{}\,\,0&{}\,\,1&{}\,\,1&{}\,\,1\\ 1&{}\,\,0&{}\,\,1&{}\,\,0&{}\,\,1&{}\,\,1&{}\,\,1\\ 1&{}\,\,1&{}\,\,0&{}\,\,0&{}\,\,1&{}\,\,1&{}\,\,1\\ 0&{}\,\,0&{}\,\,0&{}\,\,1&{}\,\,0&{}\,\,1&{}\,\,1\\ 0&{}\,\,0&{}\,\,0&{}\,\,1&{}\,\,1&{}\,\,0&{}\,\,1\\ 0&{}\,\,0&{}\,\,0&{}\,\,1&{}\,\,1&{}\,\,1&{}\,\,0\\ \end{array}\right) , Z_3=\left( \begin{array}{ccccccc} 1&{}\,\,1&{}\,\,0&{}\,\,0&{}\,\,0&{}\,\,1&{}\,\,0\\ 0&{}\,\,1&{}\,\,0&{}\,\,1&{}\,\,1&{}\,\,1&{}\,\,1\\ 1&{}\,\,0&{}\,\,0&{}\,\,1&{}\,\,1&{}\,\,1&{}\,\,1\\ 1&{}\,\,1&{}\,\,0&{}\,\,1&{}\,\,1&{}\,\,0&{}\,\,1\\ 0&{}\,\,0&{}\,\,1&{}\,\,0&{}\,\,1&{}\,\,0&{}\,\,1\\ 0&{}\,\,0&{}\,\,1&{}\,\,1&{}\,\,0&{}\,\,0&{}\,\,1\\ 0&{}\,\,0&{}\,\,1&{}\,\,1&{}\,\,1&{}\,\,0&{}\,\,0\\ \end{array}\right) ,$ and the generator matrices are given so that $X_f=\{13,14\}.$

Although we have constructed the two direct summands for the code C we have to attach them together. Let the subcode $F_\sigma (C)$ be fixed as generated by the matrix $G_2$ or $G_3.$ We have to consider all (even equivalent) possibilities for the second subcode $E_\sigma (C)$.

Let $Q_i, i=2,3$ be the subgroup of the automorphism group of the [14, 7] binary code generated by $G_i$ consisting of the automorphisms of this code that permute the first 12 coordinates (corresponding to the 5-cycle coordinates) among themselves and permute the last 2 coordinates (corresponding to the fixed point coordinates) among themselves. Let $St_i, i=2,3$ be the subgroup of the symmetric group $S_{12}$ consisting of the permutations in $Q_i$ restricted to the first 12 coordinates, ignoring the action on the fixed points. We have:

$$\begin{aligned} St_2=&\langle (1,9,4,2)(3,8)(5,11), (1,10,9,4,2,8,3)(5,7)(6,11)\rangle , \\ St_3=&\langle (1,3,9)(2,4,8)(5,10)(6,7), (1,10,2,12,3,5)(4,7,9,11,8,6)\rangle , \end{aligned}$$

$|St_2|=1344,$ and $|St_3|=1152.$

For a permutation $\tau \in S_{12}$ denote by $C_{i,j}^{\tau }$, $i=2,3,$ $j=1,\ldots ,60{,}467$ the [62, 31] self-dual code determined by the matrix $G_i$, with columns permuted by $\tau$, as a generator for $F_\sigma (C)$ and $H_j$ as a generator matrix for $E_\sigma (C)^*$. If $\tau _1$ and $\tau _2$ belong to one and the same right coset of $St_2$ (or $St_3$) in $S_{12}$, then the codes $C_{i,j}^{\tau _1}$ and $C_{i,j}^{\tau _2}$ are equivalent. Thus we can only use the right transversals $T_2$ and $T_3$ of $S_{12}$ with respect to $St_2$ and $St_3.$ We have calculated $|T_2|=356{,}400, |T_3|=415{,}800.$ After calculating all codes $C_{i,j}^{\tau }$, $i=2,3,$ $j=1,\ldots ,60{,}467$ for $\tau \in T_i,$ $i=2,3$ we summarize the results as follows.

Theorem 5

In total there are exactly 4636 inequivalent binary self-dual [62, 31, 12] codes with an automorphism of type 5-(12, 2). There exist binary self-dual [62, 31, 12] codes with weight enumerator $W_{62,2}$ for $\beta =0,1,6,11$ and 21.

Remark 3

We have checked a total of more than 46 billion codes. Computational time for this length was about a week on a 4 core 3Ghz CPU. We have the following result.

The complete information on codes obtained is listed in Table 7 for codes when $C_\pi$ is generated by $G_2$ and in Table 8 for the other case. Our results show only codes with weight enumerator $W_{62,2}.$ The codes in Table 7 all have $|\text{ Aut }(C)|=5$ that is the reason we only give their weight distribution. We note that the values $\beta =0, 1, 6, 11,$ and 21 for $W_{62,2}$ appear for the first time in the literature. Examples of codes for every new value of $\beta$ can be obtained from [7]. All self-dual [62, 31, 12] codes with $|\text{ Aut }(C)|\equiv 0 \pmod {15}$ from the paper [13] have occurred also in our results.

Table 7 Codes obtained with different $\beta$ using the matrix $G_2$

Table 8 The number of codes for different pairs $(\beta ,|\text{ Aut }(C)|)$ obtained using the matrix $G_3$

6 [64, 32, 12] binary self-dual codes with automorphism of type 5-(12, 4)

For [64, 32, 12] self-dual codes there is one possibility for a doubly-even code:

$$\begin{aligned} W_{64}=1+2976{{y}^{12}}+454{,}956{{y}^{16}}+18{,}275{,}616{{y}^{20}}+\cdots \end{aligned}$$

(4)

Such codes exist, for example in [16] they are derived from binary image of an extended Reed–Solomon code over ${\mathbb {F}}_{16}$.

The possible weight enumerators $W_{64,i}$ of extremal singly even self-dual [64, 32, 12] codes are given in [10]:

$$\begin{aligned} W_{64,1} =&1+(1312 + 16 \beta )y^{12} +(22{,}016 -64 \beta ) y^{14}+ \cdots ,\\ W_{64,2} =&1+(1312 + 16 \beta ) y^{12}+(23{,}040 - 64 \beta ) y^{14}+ \cdots , \end{aligned}$$

where $\beta$ are integers with $14 \le \beta \le 104$ for $W_{64,1}$ and $0 \le \beta \le 277$ for $W_{64,2}$. Extremal singly even self-dual codes with weight enumerator $W_{64,1}$ are known for

$$\begin{aligned} \beta \in \left\{ \begin{array}{l} 14, 16, 18, 20, 22, 24, 25, 26, 28, 29, 30, 32, \\ 34, 35, 36, 38, 39, 44, 46, 53, 59, 60, 64, 74 \end{array}\right\} \end{aligned}$$

(see [15, 17,18,19]). Extremal singly even self-dual codes with weight enumerator $W_{64,2}$ are known for

$$\begin{aligned} \beta \in \left\{ \begin{array}{l} 0,1,\ldots , 42, 44, 45, 48, 50, 51, 52, 56, 58, 64, 65, 72, \\ 80, 88, 96, 104, 108, 112, 114, 118, 120, 184 \end{array}\right\} \setminus \{31, 39\} \end{aligned}$$

(see [15, 17,18,19]).

In this case $C_\pi$ is a binary self-dual [16, 8] code. There are exactly seven such codes: five singly even $i_2\oplus 2e_7,$ $2i_2\oplus d_{12},$ $4i_2\oplus e_8,$ $8i_2,$ $2d_8$ and two doubly-even $d_{16}$ and $2e_8.$ The minimum weight $d=12$ of the code C limits the minimum weight of $C_\pi$ to $d'\ge 4$ effectively eliminating all codes with the summand $i_2.$ Using the codes $2d_8, d_{16},$ and $2e_8$ for all possible $\left( {\begin{array}{c}16\\ 4\end{array}}\right)$ splittings of $\{1,\ldots ,16\}$ into sets $X_c$ and $X_f,$ we have calculated the minimum weight of the code $F_\sigma (C).$ For a code $C_\pi$ there occur a total of 8 different generator matrices: one from $2e_8$ generating a doubly-even subcode $F_\sigma (C);$ six from $d_{16}$ with all six codes singly-even; and one doubly-even code from $2d_8.$ Denote by $G_4,\dots ,G_{11}$ the generator matrices of these 8 codes, only the matrices $G_9$ and $G_{10}$ are not in standard form. Assuming that $X_c=\{1,\ldots ,12\}$ we give the support of the rows of the matrices $G_4,\ldots ,G_{11}$ in Table 9 (for shortness the coordinates $10,11,\ldots ,16$ are denoted by the letters $a,b,\ldots g,$ respectively). We note $\pi ^{-1}(G_4)$ and $\pi ^{-1}(G_{11})$ generate doubly-even subcodes $F_\sigma (C)$ and therefore only in both those cases the [64, 32, 12] codes will be doubly-even.

Table 9 The generator matrices $G_4,\ldots ,G_{11}$

For $4\le i\le 11,$ using the double transversal $T_i,$ of $S_{12}$ with respect to the groups $St_i$ and denoting $C_{i,j}^{\tau }$ the code determined by the matrix $G_i$, with columns permuted by $\tau$, as a generator for $F_\sigma (C)$ and $H_j,$ as a generator matrix for $E_\sigma (C)^*$, we have calculated the weight distribution of all codes, except for $C_{4,j}^\tau$ and $C_{11,j}^\tau$ where the resulting [64, 32, 12] codes are doubly-even. For the codes $C_{4,j}^\tau$ and $C_{11,j}^\tau$, due to the huge computer time needed to find all codes, we have calculated only the codes for which the automorphism group of $H_j$ is not of order 5, 10, 20, and 40.

Up to equivalence we summarize our results for code with $|\text{ Aut }(C)|\ne 5$ when $\text{ gen }(C_\pi )=G_4$ and $\text{ gen }(C_\pi )=G_{11}$ in Tables 10 and 11, respectively.

Table 10 Number of doubly-even codes for different values of $|\text{ Aut }(C)|$, $\text{ gen }(C_\pi )=G_4$

Table 11 Number of doubly-even codes for different values of $|\text{ Aut }(C)|$, $\text{ gen }(C_\pi )=G_{11}$

Table 12 Generators of $St_i,$ cardinality of transversals $T_i$ and computational time for $4\le i\le 11$

Examining the singly-even [64, 32, 12] codes with an automorphism of type 5-(12, 4) we have calculated their weight distributions and we also did a check for equivalence. The cardinality of the transversals $T_4,\dots ,T_{11}$ and the computational time used to compute these cases are given in Table 12. We have checked a total of more than 530 billion codes. Computational time for this length was about 2 months on a 4 core 3Ghz CPU. We have the following result.

Theorem 6

Up to equivalence there exists exactly 6, 834, 068 binary singly-even [64, 32, 12] codes with an automorphism of type 5-(12, 4). Of these codes 1469019 and 5365049 have weight enumerator $W_{64,1}$ and $W_{64,2},$ respectively. There exist codes with $W_{64,1}$ for $\beta =19,$ 49,  and, 54,  and $W_{64,2}$ for $\beta =31,$ 39,  46,  47,  49,  54,  55,  57,  60,  62,  and 69.

Remark 4

Examples of codes for every new value of $\beta$ are listed in [7] (Tables 13, 14, 15, 16, 17, 18).

Table 13 Values of $(\beta ,|\text{ Aut }(C)|)$ for $\text{ gen }(C_\pi )=G_5$, all codes with $W_{64,2}$

Table 14 Values of $(\beta ,|\text{ Aut }(C)|)$ for $\text{ gen }(C_\pi )=G_6$, all codes with $W_{64,2}$

Table 15 Values of $(\beta ,|\text{ Aut }(C)|)$ for $\text{ gen }(C_\pi )=G_7$, all codes with $W_{64,1}$

Table 16 Values of $(\beta ,|\text{ Aut }(C)|)$ for $\text{ gen }(C_\pi )=G_8$, all codes with $W_{64,2}$

Table 17 Values of $(\beta ,|\text{ Aut }(C)|)$ for $\text{ gen }(C_\pi )=G_9$, all codes with $W_{64,2}$

Table 18 Values of $(\beta ,|\text{ Aut }(C)|)$ for $\text{ gen }(C_\pi )=G_{10}$, all codes with $W_{64,1}$

7 New [58, 29, 10] binary self-dual codes

There are two possible weight enumerators for a self-dual [58, 29, 10] code in [10]. Harada in [11] proved that indeed the first weight enumerator only occur for $\gamma =55.$ Thus we have the following enumerators:

$$\begin{aligned} W_{58,1}&=1+55y^{10}+5188y^{12}+18{,}180y^{14}+432{,}333y^{16}+\ldots ,\\ W_{58,2}&=1+(319-24\beta -2\gamma )y^{10}+(3132+152\beta +2\gamma )y^{12}+\ldots , \end{aligned}$$

where $0\le \beta \le 11$ and $0\le \gamma \le 159-12\beta .$ Codes are known with $W_{58,1}$ and with $W_{58,2}$ for [11]:

• $\beta =0$, $\gamma \in \{2m\vert m=0, \ldots ,65,68,71,79\};$

• $\beta =1$, $\gamma \in \{2m\vert m=8, \ldots , 58, 63\};$

• $\beta =2$, $\gamma \in \{2m\vert 0, 4, 6, \ldots , 55\}.$

Let C be a self-dual [60, 30, 12] code. By choosing a pair $1\le i_1<i_2\le 60$ of coordinates we can construct a new code [20]

$$\begin{aligned} C'=\{(x_1,\ldots , x_{i-1},x_{i+1},\ldots ,x_{j-1},x_{j+1},\ldots ,x_n) \vert (x_1,\ldots ,x_{60})\in C_{60,i}, x_{i_1}=x_{i_2}\}. \end{aligned}$$

It is well known that $C'$ is a self-dual code of length 58 and we say that $C'$ is obtained from C by subtracting. Since all codes we are shortening have minimum weight 12, all codewords obtained have minimum weight 10 so all $C'$ are self-dual [58, 29, 10] codes.

We start with the 315 binary self-dual [60, 30, 12] codes with an automorphism of order 5: 236 constructed in Sect. 4, and the 79 codes with an automorphism of type 5-(10, 10) from [21]. Since the minimum weight of all codes we are shortening is 12, all new codewords have minimum weight 10, so all codes $C'$ are in fact optimal self-dual [58, 29, 10] codes. By shortening for all pair $(i_1,i_2), 1\le i_1<i_2\le 60$ we obtain the following result.

Proposition 3

Up to equivalence there are exactly 53, 968 binary self-dual [58, 29, 10] codes obtained by subtracting the [60, 30, 12] self-dual code with an automorphism of type 5-(12, 0). Of these codes 189 have $W_{58,1}$ and 53, 779 have $W_{58,2}$ for 80 different pairs $(\gamma ,\beta ):$

• $\beta =0, \gamma =2m, m\in \{0,26,29,\ldots ,64, 66\};$

• $\beta =1, \gamma =2m, m\in \{39,\ldots ,55\};$

• $\beta =2, \gamma =2m, m\in \{26,28,\ldots ,51\}.$

Remark 5

For the first time in the literature we construct [58, 29, 10] codes with $W_{58,2}$ for $\beta =0,$ $\gamma =132.$ Of the three codes constructed 2 have automorphism group of 4 elements and one has $|\text{ Aut }(C)|=8$. All codes with $|\text{ Aut }(C)|\equiv 0\pmod {5}$ have an automorphism of type 5-(10, 8) an thus are known from [21]. All other codes are new. An example of a code for the parameters $\beta =0,$ $\gamma =132$ in $W_{58,2}$ is available in [7].