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.     

The existence of extremal self-dual [50,25,10] codes and quasi-symmetric 2-(49,9,6) designs

All extremal binary self-dual [50,25,10] codes with an automorphism of order 7 are enumerated. Up to equivalence, there are four such codes, three with full automorphism group of order 21, and one code with full group of order 7. The minimum weight codewords yield quasi-symmetric 2-(49,9,6) designs.Research supported in part by NSA grant MDA904-95-H-1019.     

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.     

Binary extremal self-dual codes of length 60 and related codes

We give a classification of four-circulant singly even self-dual [60, 30, d] codes for \(d=10\) and 12. These codes are used to construct extremal singly even self-dual [60, 30, 12] codes with weight enumerator for which no extremal singly even self-dual code was previously known to exist. From extremal singly even self-dual [60, 30, 12] codes, we also construct optimal singly even self-dual [58, 29, 10] codes with weight enumerator for which no optimal singly even self-dual code was previously known to exist. Finally, we give some restriction on the possible weight enumerators of certain singly even self-dual codes with shadow of minimum weight 1.     

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.      

Weight Enumerators of Double Circulant Codes and New Extremal Self-Dual Codes

In this paper it is shown that the weight enumerator of a bordered double circulant self-dual code can be obtained from those of a pure double circulant self-dual code and its shadow through a relationship between bordered and pure double circulant codes. As applications, a restriction on the weight enumerators of some extremal double circulant codes is determined and a uniqueness proof of extremal double circulant self-dual codes of length 46 is given. New extremal singly-even [44,22,8] double circulant codes are constructed. These codes have weight enumerators for which extremal codes were not previously known to exist.     

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.      

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 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.     

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.     

The [52, 26, 10] Binary Self-Dual Codes with an Automorphism of Order 7

All [52, 26, 10] binary self-dual codes with an automorphism of order 7 are enumerated. Up to equivalence, there are 499 such codes. They have two possible weight enumerators, one of which has not previously arisen.     

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.     

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.     

A quasi‐symmetric 2‐(49,9,6) design

Huffman and Tonchev discovered four non‐isomorphic quasi‐symmetric 2‐(49,9,6) designs. They arise from extremal self‐dual [50,25,10] codes with a certain weight enumerator. In this note, a new quasi‐symmetric 2‐(49,9,6) design is constructed. This is established by finding a new extremal self‐dual [50,25,10] code as a neighbor of one of the four extremal codes discovered by Huffman and Tonchev. A number of new extremal self‐dual [50,25,10] codes with other weight enumerators are also found. © 2002 Wiley Periodicals, Inc. J Combin Designs 10: 173–179, 2002; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10007     

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.     

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     

Classification of Hadamard matrices of order 44 with automorphisms of order 7

The Hadamard matrices of order 44 possessing automorphisms of order 7 are classified. The number of their equivalence classes is 384. The order of their full automorphism group is calculated. These Hadamard matrices yield 1683 nonisomorphic 3-(44,22,10) designs, 57932 nonisomorphic 2-(43,21,10) designs, and two inequivalent extremal binary self-dual doubly even codes of length 88 (one of them being new).     

Near-Extremal Formally Self-Dual Even Codes of Lengths 24 and 32

The weight enumerator of a formally self-dual even code is obtained by the Gleason theorem. Recently, Kim and Pless gave some restrictions on the possible weight enumerators of near-extremal formally self-dual even codes of length divisible by eight. In this paper, the weight enumerators for which there is a near-extremal formally self-dual even code are completely determined for lengths 24 and 32, by constructing new near-extremal formally self-dual codes. We also give a classification of near- extremal double circulant codes of lengths 24 and 32. Communicated by: P. Fitzpatrick     

Extremal self-dual codes of length 64 through neighbors and covering radii

We construct extremal singly even self-dual [64,32,12] codes with weight enumerators which were not known to be attainable. In particular, we find some codes whose shadows have minimum weight 12. By considering their doubly even neighbors, extremal doubly even self-dual [64,32,12] codes with covering radius 12 are constructed for the first time.     

Self-Dual [24, 12, 8] Quaternary Codes with a Nontrivial Automorphism of Order 3

All (Hermitian) self-dual [24, 12, 8] quaternary codes which have a non-trivial automorphism of order 3 are obtained up to equivalence. There exist exactly 205 inequivalent such codes. The codes under consideration are optimal, self-dual, and have the highest possible minimum distance for this length.