Self-dual codes with an automorphism of order 13

Using a method for constructing binary self-dual codes having an automorphism of odd prime order \begin{document}$p$\end{document} we classify, up to equivalence, all singly-even self-dual \begin{document}$[78,39,14]$\end{document} , \begin{document}$[80,40,14]$\end{document} , \begin{document}$[82,41,14],$\end{document} and \begin{document}$[84,42,14]$\end{document} codes as well as all doubly-even \begin{document}$[80,40,16]$\end{document} codes for \begin{document}$p=13$\end{document} . The results show that there are exactly 1592 inequivalent binary self-dual \begin{document}$[78,39,14]$\end{document} codes with an automorphism of type \begin{document}$13-(6,0)$\end{document} and we found 6 new values of the parameter in the weight function thus increasing more than twice the number of known values. As for binary \begin{document}$[80,40]$\end{document} self-dual codes with an automorphism of type \begin{document}$13-(6,2)$\end{document} there are 162696 singly-even self-dual codes with minimum distance 14 and 195 doubly-even such codes with minimum distance 16. We also construct many new codes of lengths 82 and 84 with minimum distance 14. Most of the constructed codes for all lengths have weight enumerators for which the existence was not known before.


Introduction
Let 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 F n q . The elements of C are called codewords, and the (Hamming) weight of a codeword v ∈ C is the number of the non-zero coordinates of v. We use 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 (denoted by gen(C)).
By O, I, J we denote the zero, identity and all-ones matrices, respectively.
For every u = (u 1 , . . . , u n ) and v = (v 1 , . . . , v n ) from F n 2 , u.v = n i=1 u i v i defines the inner product in F n 2 . The dual code of C is C ⊥ = {v ∈ F n 2 | u.v = 0, ∀ u ∈ C}. If C ⊆ C ⊥ , C is called self-orthogonal, and if C = C ⊥ , we say that C is self-dual. We call a binary code self-complementary if it contains the all-ones vector. Every binary self-dual code is self-complementary.
A 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 ≡ 2 (mod 4). Self-dual doubly-even codes exist if and only if n is a multiple of eight.
The weight enumerator W (y) of a code C is defined as W (y) = n i=0 A i y i , where A i is the number of codewords of weight i in C. We say that two linear codes C and C are permutation 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 denoted by Aut(C).
Codes achieving this bound are called extremal. A self-dual code is called optimal if it has the highest minimum weight among all self-dual codes.
Optimal self-dual codes with an automorphism of odd prime order are a well studied subject. In fact all such codes are classified up to length 50 [18]. By Gilliver and Harada [8] there exist 5 extremal double circulant doubly-even self-dual codes of length 80 with an automorphism of order 13. The orders of authomorphism groups of those codes are: 246680 for one code (P 80,1 = B 80,5 ) which is both pure and bordered double circulant; 4 codes (B 80,1 -B 80,4 ) have |Aut(C)| = 78. In [6] P. Gaborit  The extremal or optimal binary self-dual codes with an automorphism of order 13 with 4 cycles are classified in [19]. We continue the investigation of binary self-dual codes with an automorphism of order 13 with the next possible case, i.e. with 6 independent 13-cycles. We should note that St. Kapralov et al. in [14] have constructed 35 doubly-even [80,40,16] codes with an automorphism of type 13-(6, 2) but give no classification.
By the Rains bound (1) we have d ≤ 16 for binary self-dual codes of lengths 78 to 92. Let C be an optimal binary self-dual [2k, k, d] code for 39 ≤ k ≤ 46 and d = 16 or 14. We assume that the code C has an automorphism σ of order 13 with c = 6 cycles and f fixed points for 0 ≤ f = 78 − 2k ≤ 14. By [21] if f > c = 6 we have f ≥ (f −c)/2−1 k=0 d 2 k , which for d = 16 and k ≥ 1 gives f ≥ 16. Therefore we have f = 0, 2, 4 or 6 and 39 ≤ k ≤ 42.
The possible weight enumerators for extremal and optimal binary self-dual codes of lengths 72 ≤ n ≤ 100 are known from Steven Dougherty, T. Aaron Gulliver and Masaaki Harada [5]. Later in [11]  The extended quadratic residue code of length 80 is a known extremal code of this length, there are also 11 inequivalent such codes with an automorphism of order 19 [3], more codes are constructed in [14]. By the Assmus-Mattson theorem, the codewords of a fixed weight in an extremal doubly-even [80, 40, 16] code form a 3-design (see [3]). For the weight distribution of a singly-even [80,40,14] self-dual code we have found the formula: where α, β are integer parameters. One singly-even self-dual [80,40,14] code is constructed in [4] but its weight enumerator is not given. A pure double circulant such code is constricted in [8] with α = −280, β = 10.
Considering self-dual [82, 42, 14] codes, we have found codes with the following two weight enumerators: W 82,1 = 1 + (3280 + 4α)y 14 + (36244 − 4α + 128β)y 16 +(514345 − 52α − 896β)y 18 + · · · , W 82,2 = 1 + (3280 + 4α)y 14 + (36244 − 4α + 128β)y 16 where α and β are integer parameters. Pure double circulant codes with W 82,1 for α = −328, β = 0 are constructed in [5] and [8]. The [84,42,14] self-dual codes possess two weight enumerators: where α, β are integer parameters. There is one bordered double circulant code from [8] which have W 84,2 for α = 3342. Also a self-dual [84,42,14] code is constructed in [6]. This work is organized as follows. In Section 2 we will give the construction method used. In Sections 3 we classify hermitian self-dual codes of length 6 over the set of all even-weight polynomials in F 2 [x]/ x 13 − 1 . Using these codes, in Sections 4-7 we classify all optimal self-dual codes of lengths 78 ≤ n ≤ 84 with an automorphism of order 13 with 6 cycles. Remark 1. The computations were made by two of the authors independently. Both computations match exactly. One of the computations use GAP 4.8 [7] for the generation of the codes and Q-extension [2] for the code equivalence. The second computation was made with own Delphi code for code generation and Qextension for the code equivalence. The generating parameters for the matrices of the codes from Sections 3-7 are available in [17].

Construction method
Let C be a binary self-dual code of length n with an automorphism σ of odd prime order p with exactly c independent p-cycles and f = n − pc fixed points in its decomposition. We may assume that Denote the cycles of σ by Ω 1 , . . . , Ω c , and the fixed points by Ω c+1 , . . . , Ω c+f . Let . Assume that C is a self-dual code. Then the code C is a direct sum of the subcodes F σ (C) and E σ (C). The subcodes F σ (C) and E σ (C) are subspaces of dimensions c+f 2 and c(p−1)
where P is the set of even-weight polynomials in the factor ring Thus we obtain the map ϕ : E σ (C) * → P c . P is a cyclic code of length p with generator polynomial x − 1. It is known that ϕ(E σ (C) * ) is a submodule of the P-module P c [12], [20]. 20]). A binary [n, n/2] code C with an automorphism σ is self-dual if and only if the following two conditions hold: In order to classify the codes that we have obtained we need additional conditions for equivalence given by the following.

Theorem 2.3 ([21]
). 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 → x 2 in C ϕ ; (iv) a cyclic shift to each p-cycle independently.

3.
Constructing the E σ (C) * subcode By [19], 2 is a primitive root modulo 13, and hence P is a field with 2 12 elements and identity e(x) = x + · · · + x 12 . We use the element α = 1 + x + x 3 + x 5 which is a primitive element in P [21]. Denote β = α 13 an element of multiplicative order 315 in P. We can write After Gaussian elimination we can take the generator matrix to be in the form G = (I|Z), where Z is a 3 × 3 matrix over P. Using Theorem 2.3 we can transform the matrix Z to the following , or some of the elements in Z are zeroes. Using the orthogonal condition (2) and checking that d = 16 we calculated all possible inequivalent choices of the first row of Z and found 1676 triples (i 1 , i 2 , i 3 ). Next, we added the second row of Z and we obtained 4086196 different 2 × 3 submatrices. Finally, after adding the last row we obtained exactly 322103 inequivalent codes with minimum distance d = 16.
Theorem 3.1. There are exactly 322103 inequivalent codes C ϕ of length 6 over the set P of all even-weight polynomials in The number of inequivalent codes E σ (C) * sorted by A 16 (the number of different codewords of weight 16) are given in Table 1. The order of the automorphism groups of the codes that we have obtained are listed in Table 2. Assume that C is a [78, 39, 14] self-dual code with an automorphism of type 13 − (6, 0). By Theorem 2.2, C π is a binary self-dual [6,3,2] code. There is a unique such code: 3i 2 ([13]) with generator matrix G 1 = (I 3 |I 3 ). By Theorem 2.1 C is a direct sum of F σ (C) and E σ (C). We fix the generator matrix of E σ (C) to be the generator matrix of one of the codes from Theorem 3.1. For all permutations τ ∈ S 6 we consider the generator matrix of C π to be τ (G 1 ). We summarize the results in the following.  Table 3 where the new values are marked in bold. One of the three codes with the pair (β, |Aut(C)|) = (−78, 78) is the code C 78,1 from [9]. 5. Classification of doubly-even [80, 40, 16] and singly-even [80,40,14] self-dual codes with an automorphism of type 13 − (6, 2) Let C be a [80, 40, d] self-dual code with an automorphism of type 13 − (6, 3). By Theorem 2.2, C π is a binary self-dual [8, 4, ≥ 2] code. There are two such codes ( [13]): the singly-even 4i 2 and the doubly-even h 8 . Since we need to choose 2 out of 8 coordinate positions for the set X f of fixed points, in the case 4i 2 , in order to       Table 5 we give the number of codes for the different pairs (α, |Aut(C)|). We note that for all these values in W 80,2 there were previously no known codes. M. Harada and A. Munemasa in [10] have established the weight enumerators of a putative s-extremal singly-even selfdual [80,40,14] code, none of the codes we have obtained is s-extremal (W 80,2 for α = β = 0). Let C be a [82, 41, 14] self-dual code with an automorphism of type 13 − (6, 4). By Theorem 2.2 C π is a binary self-dual [10, 5, ≥ 2] code. There are two such codes ( [13]): 5i 2 and i 2 + h 8 . Checking the possible arrangements of the 10 coordinate positions of both codes into subsets X c and X f , we found 3 possible generator matrices for C π : where G 4 is equivalent to 5i 2 and the other two are equivalent to i 2 + h 8 . We fix E σ (C) and consider all permutations τ ∈ S 6 acting on the cyclic points in G i , i = 4, 5, 6. • gen(C π ) = G 4 : 604992 codes with weight enumerator W 82,2 ; • gen(C π ) = G 5 : 164338 codes with weight enumerator W 82,2 ; • gen(C π ) = G 6 : 50989 codes with weight enumerator W 82,1 .
When C π = G 4 the codes have weight enumerator W 82,2 for different values of α for β = 0, 13, and 26. All codes that we have obtained with β = 0 are listed in Table 6, with β = 13 in Table 7, and there is a unique code with β = 26, α = −364 and an automorphism group with 78 elements.
When C π = G 5 the codes have weight enumerator W 82,2 for different values of α for β = 0, and 13. All codes that we have obtained with β = 0 are listed in Table  8 and with β = 13 in Table 9.
We fix E σ (C) and consider all permutations τ ∈ S 6 acting on the cyclic points in G i , i = 7, . . . , 10.
When C π = G 7 the codes have weight enumerator W 84,2 for different values of α. We give the number of codes with different pairs (α, |Aut(C)|) that we have obtained in Table 10.
In the case of C π = G 9 the codes have weight enumerator W 84,2 for different values of α. We give the number of codes with different pairs (α, |Aut(C)|) that we have obtained in Table 11.