Doubly-even Self-dual Codes Of Length Research Articles

Mutually disjoint t-designs and t-SEEDs from extremal doubly-even self-dual codes

It is known that extremal doubly-even self-dual codes of length $$n\equiv 8$$ n ? 8 or $$0\ (\mathrm {mod}\ 24)$$ 0 ( mod 24 ) yield 3- or 5-designs respectively. In this paper, by using the generator matrices of bordered double circulant doubly-even self-dual codes, we give 3-(n, k; m)-SEEDs with (n, k, m) $$\in \{(32,8,5), (56,12,9), (56,16,9), (56,24,9), (80,16,52)\}$$ ? { ( 32 , 8 , 5 ) , ( 56 , 12 , 9 ) , ( 56 , 16 , 9 ) , ( 56 , 24 , 9 ) , ( 80 , 16 , 52 ) } . With the aid of computer, we obtain 22 generator matrices of bordered double circulant doubly-even self-dual codes of length 48, which enable us to get 506 mutually disjoint 5-(48, k, $$\lambda $$ ? ) designs for (k, $$\lambda $$ ? )=(12, 8),(16, 1356),(20, 36176). Moreover, this implies 5-(48, k; 506)-SEEDs for $$k=12, 16, 20, 24$$ k = 12 , 16 , 20 , 24 .

On Extremal Binary Doubly-Even Self-Dual Codes of Length 88*

In this paper we present 35 new extremal binary self-dual doubly-even codes of length 88. Their inequivalence is established by invariants. Moreover, a construction of a binary self-dual [88, 44, 16] code, having an automorphism of order 21, is given.

3-(2 m , 2m–1, 2m–2–1) design for first order Reed-Muller code R(1, m)

A binary linear code in with dimension k and minimum distance d is called an [n, k, d] code. For each positive integer m, the first order Reed-Muller code R(1, m) is a linear code [2 m , m+1, 2 m–1]. When m=3, an extremal doubly-even self-dual code of length 23=8 exists. In this paper, it is shown that the first order Reed-Muller code R(1, m) holds a 3-(2 m , 2 m–1, 2 m–2–1) design for m≥3. It is also shown that Steiner system holds for m=3.

Extremal doubly-even self-dual codes from Hadamard matrices

In this paper, new extremal doubly-even self-dual codes are obtained by using some newly constructed Hadamard matrices of various orders. In particular, many inequivalent extremal doubly-even self-dual codes of length 40, 48, 56 and 64 are presented. These codes derived from Hadamard matrices of the same order.

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,...

Remark on a 5-Design Related to a Putative Extremal Doubly-Even Self-Dual [96, 48, 20] Code

In this note, it is shown that if there is a self-orthogonal 5-(96,20,816) design, then the rows of its block-point incidence matrix generate an extremal doubly-even self-dual code of length 96. In other words, a putative extremal doubly-even self-dual code of length 96 is generated by the codewords of minimum weight.

A Characterization of Designs Related to an Extremal Doubly-Even Self-Dual Code of Length 48

The uniqueness of a binary doubly-even self-dual [48, 24, 12] code is used to prove that a self-orthogonal 5-(48, 12, 8) design, as well as some of its derived and residual designs, including a quasi-symmetric 2-(45, 9, 8) design, are all unique up to isomorphism.

Doubly-even self-dual code of length 96

We prove that 23, 11, and 7 do not divide the order of the automorphism group of a binary [96, 48, 20] doubly-even self-dual code. We construct 25 new inequivalent binary [96, 48, 16] doubly-even self-dual codes via an automorphism of order 23.

The mass of extremal doubly-even self-dual codes of length 40

We determine the mass of extremal doubly-even self-dual binary codes of length 40. It follows that there are at least 12579 such codes.

An improved upper bound on the minimum distance of doubly-even self-dual codes

We derive a new upper bound on the minimum distance d of doubly-even self-dual codes of length n. Asymptotically, for n growing, it gives lim/sub n/spl rarr//spl infin// sup d/n/spl les/(5-5/sup 3/4/)/10<0.165630, thus improving on the Mallows-Odlyzko-Sloane bound of 1/6 and our recent bound of 0.166315.

New extremal self-dual codes of length 68

We give two computational results on binary self-dual codes. A number of extremal singly-even self-dual [68, 34, 12] codes with weight enumerators not known to exist, are constructed. For k/spl les/10, the relationship between extremal doubly-even self-dual codes of length 24 k, and certain singly-even self-dual codes of length 24 k-2 is presented.

Linear programming bounds for doubly-even self-dual codes

Using a variant of the linear programming method we derive a new upper bound on the minimum distance d of doubly-even self-dual codes of length n. Asymptotically, for n growing, it gives d/n/spl les/0.166315/spl middot//spl middot//spl middot/+o(1), thus improving on the Mallows-Odlyzko-Sloane bound of 1/6. To establish this, we prove that in any doubly even-self-dual code the distance distribution is asymptotically upper-bounded by the corresponding normalized binomial distribution in a certain interval.

The binary self-dual codes of length up to 32: A revised enumeration

This paper presents a revised enumeration of the binary self-dual codes of length up to 32 given by Conway and Pless in 1978–1980. The list of 85 doubly-even self-dual codes of length 32 is essentially correct, but several of their descriptions need amending. The principal change is that there are 731 (not 664) inequivalent self-dual codes of length 30. Furthermore, there are three (not two) [28, 14, 6] and 13 (not eight) [30, 15, 6] self-dual codes. Some additional information is provided about the self-dual codes of length less than 32.

Self-dual codes and Hadamard matrices

A general method unifying known constructions of binary self-orthogonal codes from combinatorial designs and Hadamard matrices is described. As an application more than 80 inequivalent extremal doubly-even self-dual codes of length 40, 56 and 64 are constructed from Hadamard matrices of order 20 and 28 and symmetric 2-(31,10,3) designs. Many of these codes do not admit nontrivial automorphisms of odd orders, and there are codes with trivial automorphism groups.