Extremal Self-dual Codes Of Length Research Articles

On some self-dual codes and unimodular lattices in dimension 48

In this paper, binary extremal self-dual codes of length 48 and extremal unimodular lattices in dimension 48 are studied through their shadows and neighbors. We relate an extremal singly even self-dual [48, 24, 10] code whose shadow has minimum weight 4 to an extremal doubly even self-dual [48, 24, 12] code. It is also shown that an extremal odd unimodular lattice in dimension 48 whose shadow has minimum norm 2 relates to an extremal even unimodular lattice. Extremal singly even self-dual [48, 24, 10] codes with shadows of minimum weight 8 and extremal odd unimodular lattice in dimension 48 with shadows of minimum norm 4 are investigated.

On the Covering Radius of Ternary Extremal Self-Dual Codes

In this paper, we investigate the covering radius of ternary extremal self-dual codes. The covering radii of all ternary extremal self-dual codes of lengths up to 20 were previously known. The complete coset weight distributions of the two inequivalent extremal self-dual codes of length 24 are determined. As a consequence, it is shown that every extremal ternary self-dual code of length up to 24 has covering radius which meets the Delsarte bound. The first example of a ternary extremal self-dual code with covering radius which does not meet the Delsarte bound is also found. It is worth mentioning that the found code is of length 32.

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+224y8+872y10+·. A Type I code of length 44 with this weight enumerator was not known to exist previously.

New extremal self-dual codes of length 62 and related extremal self-dual codes

New extremal self-dual codes of length 62 are constructed with weight enumerators of three different types. Two of these types were not represented by any known code up till now. All these codes possess an automorphism of order 15. Some of them are used to construct extremal self-dual codes of length 60 by the method of subtracting. By additional subtracting, an extremal self-dual [58, 29, 10] code was obtained having a weight enumerator which does not correspond to any code known so far.

On Extremal Additive 4 Codes of Length 10 to 18

Abstract In this paper we classify the 19 even extremal self-dual additive codes of length 10 and we classify under a restrictive hypothesis 490 even extremal self-dual codes of length 14. Under the same restrictive hypothesis we also give a partial result on the uniqueness of the even self-dual additive F 4-linear [18,9,8] code.

New extremal self-dual codes of lengths 36, 38, and 58

We construct new extremal self-dual binary codes of lengths 36, 38, and 58. We show that there are at least 14 inequivalent extremal self-dual [38,18,8] codes and that there are at least 368 inequivalent extremal self-dual [38,19,8] codes. For length 58, we construct 11 extremal self-dual [58,29,10] codes whose weight enumerators were previously unknown.

A method for constructing self-dual codes with an automorphism of order 2

In this paper, we investigate binary self-dual codes with an automorphism of order 2 with c cycles and f fixed points. A method for constructing such codes using self-orthogonal codes of length c and self-dual codes of length f is presented. We apply this method to construct extremal self-dual codes of lengths 40, 42, 44, 52, 54, and 58. Some of them have weight enumerators for which self-dual codes were previously not known to exist. We prove that there do not exist self-dual [50, 25, 10] and [96, 48, 20] codes with an automorphism of order 2 with f fixed points for f>0 in their automorphism groups.

Construction of an extremal self-dual code of length 62

We construct an extremal singly-even self-dual [62, 31, 12] code. The existence of such a self-dual code was not previously known. Combined with known results, the above code also determines all lengths for which there are self-dual codes with minimum weight 12.

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.

New extremal self-dual codes of lengths 36 and 38

There are two distinct weight enumerators for extremal singly-even self-dual [36, 18, 8] codes. For each weight enumerator, only one extremal self-dual code is previously known. In this correspondence, we construct several new extremal singly-even self dual [36, 18, 8] codes for each weight enumerator. We also construct a number of new extremal singly-even self-dual [38, 19, 8] codes.

Extremal self-dual codes of lengths 66 and 68

New extremal self-dual codes of dimensions 33 and 34 are obtained.

New extremal self-dual codes of lengths 42 and 44

All extremal binary self-dual codes of lengths 42 and 44 which have an automorphism of order 5 with eight independent cycles are obtained up to equivalence. There are 109 inequivalent [42, 21, 8] codes with such an automorphism. All [44, 22, 8] codes that are obtained have 29 different weight enumerators.

Weight enumerators of extremal singly-even [60,30,12] codes

Conway and Sloane (1990) have listed the possible weight enumerators for extremal self-dual codes up to length 72. In this correspondence, we construct extremal singly-even self-dual [60,30,12] codes whose weight enumerator does not appear in this list. In addition, we present the possible weight enumerators for extremal self-dual codes of length 60.

Soft decoding performance of extremal self-dual codes

Performance of soft decoded extremal self-dual codes of lengths 8 to 72 are obtained over the Gaussian channel. The results indicate that for decoded bit-error rates below 10-3, which is the main region of interest for coding application, substantial coding gains can be obtained by using extremal self-dual codes.