Binary self-dual codes with automorphisms of order 23

The only example of a binary doubly-even self-dual [120,60,20] code was found in 2005 by Gaborit et al. (IEEE Trans Inform theory 51, 402–407 2005). In this work we present 25 new binary doubly-even self-dual [120,60,20] codes having an automorphism of order 23. Moreover we list 7 self-dual [116,58,18] codes, 30 singly-even self-dual [96,48,16] codes and 20 extremal self-dual [92,46,16] codes. All codes are new and present different weight enumerators.      

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.     

Extremal Self-Dual [50, 25, 10] Codes with Automorphisms of Order 3 and Quasi-Symmetric 2-(49, 9,6) Designs

Five non-isomorphic quasi-symmetric 2-(49, 9, 6) designs are known. They arise from extremal self-dual [50, 25, 10] codes with a certain weight enumerator. Four of them have an automorphism of order 3 fixing two points. In this paper, it is shown that there are exactly 48 inequivalent extremal self-dual [50, 25, 10] code with this weight enumerator and an automorphism of order 3 fixing two points. 44 new quasi-symmetric 2-(49, 9, 6) designs with an automorphism of order 3 are constructed from these codes.     

The binary extremal self-dual codes of lengths 38 and 40

We classify the extremal self-dual codes of lengths 38 or 40 having an automorphism of order 3 with six independent 3-cycles, 10 independent 3-cycles, or 12 independent 3-cycles. In this way we complete the classification of binary extremal self-dual codes of length up to 48 having automorphism of odd prime order.     

<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.      

Negacyclic self-dual codes over finite chain rings

In this article, we study negacyclic self-dual codes of length n over a finite chain ring R when the characteristic p of the residue field [`(R)]{\bar{R}} and the length n are relatively prime. We give necessary and sufficient conditions for the existence of (nontrivial) negacyclic self-dual codes over a finite chain ring. As an application, we construct negacyclic MDR self-dual codes over GR(p ^t , m) of length p ^m  + 1.     

Construction of MDS self-dual codes over Galois rings

The purpose of this paper is to construct nontrivial MDS self-dual codes over Galois rings. We consider a building-up construction of self-dual codes over Galois rings as a GF(q)-analogue of (Kim and Lee, J Combin Theory ser A, 105:79–95). We give a necessary and sufficient condition on which the building-up construction holds. We construct MDS self-dual codes of lengths up to 8 over GR(3²,2), GR(3³,2) and GR(3⁴,2), and near-MDS self-dual codes of length 10 over these rings. In a similar manner, over GR(5²,2), GR(5³,2) and GR(7²,2), we construct MDS self-dual codes of lengths up to 10 and near-MDS self-dual codes of length 12. Furthermore, over GR(11²,2) we have MDS self-dual codes of lengths up to 12.      

Codes Over the p-adic Integers

We study codes over the p-adic integers and correct errors in the existing literature. We show that MDS codes exist over the p-adics for all lengths, ranks and p. We show that self-dual codes exist over the 2-adics if and only if the length is a multiple of 8 and that self-dual codes exist over the p-adics with p odd if and only if the length is 0 (mod 4) for p ≡ 3 (mod 4) and 0 (mod 2) for p ≡ 1 (mod 4).     

On circulant self-dual codes over small fields

We construct self-dual codes over small fields with q = 3, 4, 5, 7, 8, 9 of moderate length with long cycles in the automorphism group. With few exceptions, the codes achieve or improve the known lower bounds on the minimum distance of self-dual codes.      

On the Classification of Self-Dual Codes over 𝔽5

 In this paper, we give the classification of self-dual 𝔽₅-codes of lengths 14 and 16. Up to equivalence, there are 53 and 535 such codes, respectively. It is also shown that there is no self-dual [18, 9, 8] code over 𝔽₅. Received: June 18, 2001 Final version received: April 9, 2002 RID="*" ID="*" Supported in part by the Academy of Finland under grants 44517 and 100500     

Singly-Even Self-Dual Codes of Length 40

All singly-even self-dual [40,20,8] binary codes which have an automorphism of prime order are obtained up to equivalence. There are two inequivalent codes with an automorphism of order 7 and 37 inequivalent codes with an automorphism of order 5. These codes have highest possible minimal distance and some of them are the first known codes with weight enumerators prescribed by Conway and Sloane.     

MDS codes over finite principal ideal rings

The purpose of this paper is to study codes over finite principal ideal rings. To do this, we begin with codes over finite chain rings as a natural generalization of codes over Galois rings GR(p ^e , l) (including ). We give sufficient conditions on the existence of MDS codes over finite chain rings and on the existence of self-dual codes over finite chain rings. We also construct MDS self-dual codes over Galois rings GF(2 ^e , l) of length n = 2 ^l for any a ≥ 1 and l ≥ 2. Torsion codes over residue fields of finite chain rings are introduced, and some of their properties are derived. Finally, we describe MDS codes and self-dual codes over finite principal ideal rings by examining codes over their component chain rings, via a generalized Chinese remainder theorem.      

The extremal codes of lengths 76 with an automorphism of order 19

We find all extremal [76,38,14] binary self-dual codes having automorphism of order 19. There are three inequivalent such codes. One of them was previously known. The other two are new. These codes are the shortest known self-dual codes of minimal weight 14 as well as the best-known linear codes of that length and dimension.     

New Optimal Self-Dual Codes over GF(7)

In this paper, double circulant self-dual codes over GF(7) are presented, including [12,6,6] codes which are new optimal codes. It is shown that the supports of the codewords of weights 9, 10 and 11 in double circulant [20,10,9] codes form 3-designs. For larger lengths, some good self-dual codes are constructed from weighing matrices. Received: June 24, 1996 / Revised: February 28, 1997     

On the Automorphisms of Order 2 with Fixed Points for the Extremal Self-Dual Codes of Length 24"m

It is shown that an extremal self-dual code of length 24">m may have an automorphism of order 2 with fixed points only for ">m = 1,3, or 5. We prove that no self-dual [72, 36, 16] code has such an automorphism in its automorphism group.     

New Extremal Type I Codes of Lengths 40, 42, and 44

It is known that it is possible to construct a generator matrix for a self-dual code of length 2n+2 from a generator matrix of a self-dual code of length 2n. With the aid of a computer, we construct new extremal Type I codes of lengths 40, 42, and 44 from extremal self-dual codes of lengths 38, 40, and 42 respectively. Among them are seven extremal Type I codes of length 44 whose weight enumerator is 1+224y ⁸+872y ¹⁰+·. A Type I code of length 44 with this weight enumerator was not known to exist previously.     

Improved lower bounds on sizes of single-error correcting codes

A construction of codes of length n = q + 1 and minimum Hamming distance 3 over is given. Substitution of the derived codes into a concatenation construction yields nonlinear binary single-error correcting codes with better than known parameters. In particular, new binary single-error correcting codes having more codewords than the best previously known in the range n ≤ 512 are obtained for the lengths 64–66, 128–133, 256–262, and 512.     

On the Binary Self-Dual Codes with an Automorphism of Order 2

Methods to design binary self-dual codes with an automorphism of order two without fixed points are presented. New extremal self-dual [40,20,8], [42,21,8],[44,22,8] and [64,32,12] codes with previously not known weight enumerators are constructed.     

Classification of Extremal Double Circulant Self-Dual Codes of Lengths 64 to 72

Recently extremal double circulant self-dual codes have been classified for lengths n ≤ 62. In this paper, a complete classification of extremal double circulant self-dual codes of lengths 64 to 72 is presented. Almost all of the extremal double circulant singly-even codes given have weight enumerators for which extremal codes were not previously known to exist.     

The classification of self-dual modular codes

A classification method of self-dual codes over Zm is given. If m=rs with relatively prime integers r and s, then the classification can be accomplished by double coset decompositions of Sn by automorphism groups of self-dual codes over Zr and Zs. We classify self-dual codes of length 4 over Zp for all primes p in terms of their automorphism groups and then apply our method to classify self-dual codes over Zm for arbitrary integer m. Self-dual codes of length 8 are also classified over Zpq for p,q=2,3,5,7.