The aim of this work is to study the structure and properties of the binary LCD codes having an automorphism of odd prime order and to present a method for their construction.

Two algorithms for the classification of linear codes over finite fields are presented. One of the algorithms is based on canonical augmentation and the other one on lattice point enumeration. New classification results over fields with 2, 3 and 4 elements are obtained.

An efficient algorithm for classification of binary self-dual codes is presented. As an application, a complete classification of the self-dual codes of length 38 is given.

In this paper, extremal singly even self-dual codes with minimal shadow are investigated. Nonexistence of such codes for particular parameters is proved. By a result of Rains, the length of extremal singly even self-dual codes is bounded. Explicit bounds are given in case the shadow is minimal.

LetCbe a binary extremal self-dual code of lengthn¿ 48. We prove that for each¿ ¿ Aut(C) of prime orderp¿ 5 the number of fixed points in the permutation action on the coordinate positions is bounded by the number ofp-cycles. It turns out that large primesp, i.e.,n-psmall, seem to occur in|Aut(C)| very rarely. Examples are the extended quadratic residue codes. We further prove that doubly even extended quadratic residue codes of lengthn=p+ 1 are extremal only in the casesn=8, 24, 32, 48, 80, and 104.

In this correspondence, we prove that the automorphism group of a putative binary self-dual doubly even [72,36,16] code is solvable. Moreover, its order is 5,7,10,14,56, or a divisor of 72

We describe a method for constructing binary self-dual codes having an automorphism of order p/sup 2/ for an odd prime p. Using this method, we classify the optimal self-dual codes of lengths 36 /spl les/ n /spl les/ 44 and n = 54, having an automorphism of order 9. We obtain all self-dual (56,28,12),(58,29,10), and (60,30,12) codes having an automorphism of order 9 without cycles of length 3. Some of the constructed codes of lengths 54,58, and 60 have weight enumerators for which the existence of codes was not known before.

We prove that a putative doubly-even (72,36,16) code has no automorphism of order 3 with fixed points. Moreover, such a code has no automorphism of an odd order greater than 7. Eventually, we prove that 25 and 49 do not divide the order of its automorphism group.

Dougherty, Gaborit, Harada, Munemasa and Sole (see ibid., vol.45, p.2345-60, 1999) have previously given an upper bound on the minimum Lee weight of a type IV self-dual Z/sub 4/-code, using a similar bound for the minimum distance of binary doubly even self-dual codes. We improve their bound, finding that the minimum Lee weight of a type IV self-dual Z/sub 4/-code of length n is at most 4[n/12], except when n=4, and n=8 when the bound is 4, and n=16 when the bound is 8. We prove that the extremal binary doubly even self-dual codes of length n/spl ges/24, n/spl ne/32 are not Z/sub 4/-linear. We classify type IV-I codes of length 16. We prove that all type IV codes of length 24 have minimum Lee weight 4 and minimum Hamming weight 2, and the Euclidean-optimal type IV-I codes of this length have minimum Euclidean weight 8.

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.

We construct extremal self-dual codes with lengths 44, 50, 54, and 58. They have weight enumerators for which extremal codes were previously not known to exist. Two methods are used for constructing the codes using self dual codes of same or smaller length. To obtain the codes we use a combinatorial optimization search.

A method to design binary self-dual codes with an automorphism of order two without fixed points is presented. Extremal self-dual codes with lengths 40, 42, 44, 54, 58, 68 are constructed. Many of them have weight enumerators for which extremal codes were previously not known to exist.

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.