On binary self-dual codes of lengths 60, 62, 64 and 66 having an automorphism of order 9

A method for constructing binary self-dual codes having an automorphism of order p ² for an odd prime p is presented in (S. Bouyuklieva et al. IEEE. Trans. Inform. Theory, 51, 3678–3686, 2005). Using this method, we investigate the optimal self-dual codes of lengths 60 ≤ n ≤ 66 having an automorphism of order 9 with six 9-cycles, t cycles of length 3 and f fixed points. We classify all self-dual [60,30,12] and [62,31,12] codes possessing such an automorphism, and we construct many doubly-even [64,32,12] and singly-even [66,33,12] codes. Some of the constructed codes of lengths 62 and 66 are with weight enumerators for which the existence of codes was not known until now.      

Existence of New Extremal Doubly-Even Codes and Extremal Singly-Even Codes

Recently the author and Kimura have considered a construction of doubly-even codes from a given doubly-even code. In this note, we show that the restricutoion of doubly-even can be removed in the above construction. As an application, at least 137 inequivalent extremal doubly-even [56,28,12] codes and at least 1000 inequivalent extremal doubly-even [40,20,8] codes are constructed from known self-dual codes. The existence of new extremal singly-even codes is also described.     

Construction of Extremal Type II Codes over ℤ4

In this paper, we give a pseudo-random method to construct extremal Type II codes overℤ₄ . As an application, we give a number of new extremal Type II codes of lengths 24, 32 and 40, constructed from some extremal doubly-even self-dual binary codes. The extremal Type II codes of length 24 have the property that the supports of the codewords of Hamming weight 10 form 5−(24,10,36) designs. It is also shown that every extremal doubly-even self-dual binary code of length 32 can be considered as the residual code of an extremal Type II code over ℤ₄.     

On the classification of the extremal self-dual codes over small fields with 2-transitive automorphism groups

There are seven binary extremal self-dual doubly-even codes which are known to have a 2-transitive automorphism group. Using representation theoretical methods we show that there are no other such codes, except possibly for length n = 1024. We also classify all extremal ternary self-dual and quaternary Hermitian self-dual codes.     

Cocyclic Codes of Length 40

There are 5 groups of order 20. This paper reports on the search for binary self-dual codes of length 40, cocyclic over any one of the first four groups, using cocyclic Hadamard matrices and the [I, A] construction. The fifth group is not investigated here. A total of 28 classes of extremal cocyclic self-dual codes were found—27 of these are doubly-even and one singly-even. The majority of these classes arise from the dihedral-cocyclic Hadamard matrices. There is also a class of dihedral-cocyclic Hadamard matrices which gives a large collection of [40, 20] codes with only one codeword of length 4.     

New extremal doubly-even [64, 32, 12] codes

In this paper, we consider a general construction of doubly-even self-dual codes. From three symmetric 2-(31, 10, 3) designs, we construct at least 3228 inequivalent extremal doubly-even [64, 32, 12] codes. These codes are distinguished by their K-matrices.     

New constructions of optimal self-dual binary codes of length 54

The quaternary Hermitian self-dual [18,9,6]₄ codes are classified and used to construct new binary self-dual [54,27,10]₂ codes. All self-dual [54,27,10]₂ codes obtained have automorphisms of order 3, and six of their weight enumerators have not been previously encountered.      

New binary self-dual codes of lengths 50–60

Using a method for constructing binary self-dual codes with an automorphism of odd prime order \(p\) , we give a full classification of all optimal binary self-dual \([50+2t,25+t]\) codes having an automorphism of order 5 for \(t=0,\dots ,5\) . As a consequence, we determine the weight enumerators for which there is an optimal binary self-dual \([52, 26, 10]\) code. Some of the constructed codes for lengths 52, 54, 58, and 60 have new values for the parameter in their weight enumerator. We also construct more than 3,000 new doubly-even \([56,28,12]\) self-dual codes.     

2-(31,15,7), 2-(35,17,8) and 2-(36,15,6) designs with automorphisms of odd prime order,and their related Hadamard matrices and codes

We present the full classification of Hadamard 2-(31,15,7), Hadamard 2-(35, 17,8) and Menon 2-(36,15,6) designs with automorphisms of odd prime order. We also give partial classifications of such designs with automorphisms of order 2. These classifications lead to related Hadamard matrices and self-dual codes. We found 76166 Hadamard matrices of order 32 and 38332 Hadamard matrices of order 36, arising from the classified designs. Remarkably, all constructed Hadamard matrices of order 36 are Hadamard equivalent to a regular Hadamard matrix. From our constructed designs, we obtained 37352 doubly-even [72,36,12] codes, which are the best known self-dual codes of this length until now.      

Self-Orthogonal 3-(56,12,65) Designs and Extremal Doubly-Even Self-Dual Codes of Length 56

In this paper, we show that the code generated by the rows of a block-point incidence matrix of a self-orthogonal 3-(56,12,65) design is a doubly-even self-dual code of length 56. As a consequence, it is shown that an extremal doubly-even self-dual code of length 56 is generated by the codewords of minimum weight. We also demonstrate that there are more than one thousand inequivalent extremal doubly-even self-dual [56,28,12] codes. This result shows that there are more than one thousand non-isomorphic self-orthogonal 3-(56,12,65) designs. AMS Classification: 94B05, 05B05     

On the enumeration of self-dual codes

We give the complete classification of all binary, self-dual, doubly-even (32, 16) codes. There are 85 non-equivalent, self-dual, doubly-even (32, 16) codes. Five of these have minimum weight 8, namely, a quadratic residue code and a Reed-Muller code, and three new codes. A set of generators is given for a code in each equivalence class together with its entire weight distribution and the order of its entire group with other information facilitating the computation of permutation generators. From this list it is possible to identify all self-dual codes of length less than 32 and the numbers of these are included.     

Binary images of some self-dual codes over GF(2m) with respect to trace-orthogonal basis

The note deals with a class of self-dual binary codes containing external doubly-even codes for lengths 8, 16, 24, 32, 40 and 64. For most of them, our construction also provides a simple method for calculating the distance of the code. To our knowledge, this is the first example of a (64, 32, 12) extermal doubly-even self-dual code.     

Computational results of duadic double circulant codes

Quadratic residue codes have been one of the most important classes of algebraic codes. They have been generalized into duadic codes and quadratic double circulant codes. In this paper we introduce a new subclass of double circulant codes, called duadic double circulant codes, which is a generalization of quadratic double circulant codes for prime lengths. This class generates optimal self-dual codes, optimal linear codes, and linear codes with the best known parameters in a systematic way. We describe a method to construct duadic double circulant codes using 4-cyclotomic cosets and give certain duadic double circulant codes over $\mathbb{F}_{2}$ , $\mathbb{F}_{3}$ , $\mathbb{F}_{4}$ , $\mathbb{F}_{5}$ , and $\mathbb{F}_{7}$ . In particular, we find a new ternary self-dual [76,38,18] code and easily rediscover optimal binary self-dual codes with parameters [66,33,12], [68,34,12], [86,43,16], and [88,44,16] as well as a formally self-dual binary [82,41,14] code.     

On the classification and enumeration of self-dual codes

A complete classification is given of all [22, 11] and [24, 12] binary self-dual codes. For each code we give the order of its group, the number of codes equivalent to it, and its weight distribution. There is a unique [24, 12, 6] self-dual code. Several theorems on the enumeration of self-dual codes are used, including formulas for the number of such codes with minimum distance ? 4, and for the sum of the weight enumerators of all such codes of length n. Selforthogonal codes which are generated by code words of weight 4 are completely characterized.     

<Emphasis Type="Italic">G</Emphasis>-projectable and Λ-projectable binary linear block codes

The binary [24,12,8] Golay code has projection O onto the quaternary [6,3,4] hexacode [9] and the [32,16,8] Reed-Muller code has projection E onto the quaternary self-dual [8,4,4] code [6]. Projection E was extended to projection G in [8]. In this paper we introduce a projection, to be called projection Λ, that covers projections O, E and G. We characterise G-projectable self-dual codes and Λ-projectable codes. Explicit methods for constructing codes having G and Λ projections are given and several so constructed codes that have best known optimal parameters are introduced.      

On the Classification of Extremal Even Formally Self-Dual Codes

Bachoc bachoc has recently introduced harmonic polynomials for binary codes. Computing these for extremal even formally self-dual codes of length 12, she found intersection numbers for such codes and showed that there are exactly three inequivalent [12,6,4] even formally self-dual codes, exactly one of which is self-dual. We prove a new theorem which gives a generator matrix for formally self-dual codes. Using the Bachoc polynomials we can obtain the intersection numbers for extremal even formally self-dual codes of length 14. These same numbers can also be obtained from the generator matrix. We show that there are precisely ten inequivalent [14,7,4] even formally self-dual codes, only one of which is self-dual.     

New optimal [52, 26, 10] self-dual codes

We classify up to equivalence all optimal binary self-dual [52, 26, 10] codes having an automorphism of order 3 with 10 fixed points. We achieve this using a method for constructing self-dual codes via an automorphism of odd prime order. We study also codes with an automorphism of order 3 with 4 fixed points. Some of the constructed codes have new values β = 8, 9, and 12 for the parameter in their weight enumerator.     

Codes with given distances

One of the main results says that ifC is a binary linear code of length 4t and of dimension greater than 2t, thenC contains a word of weight 2t and this bound is best possible. Several results of similar flavor are established both for linear and non-linear codes. For the proof a lemma introducing the binormal forms of binary matrices is needed. The results are applied to determine the coset chromatic number of Hadamard graphs, to solve a problem of Galvin and to give a short proof of a theorem of Gleason on self-dual doubly-even codes.