We investigate right ideals as codes in twisted group algebras. Such codes are called twisted group codes. It turns out that many interesting codes belong to this class; for instance, the ternary extended Golay code, Hamming codes and constacyclic codes. In particular we characterize all linear codes which are twisted group codes in terms of their automorphism group.

Let$C$be a binary self-dual code with an automorphism$g$of order$2p$, where$p$is an odd prime, such that$g^{p}$is a fixed point free involution. If$C$is extremal of length a multiple of 24, all the involutions are fixed point free, except the Golay Code and eventually putative codes of length 120. Connecting module theoretical properties of a self-dual code$C$with coding theoretical ones of the subcode$C(g^{p})$which consists of the set of fixed points of$g^{p}$, we prove that$C$is a projective$ {\BBF }_{2}\langle g \rangle $-module if and only if a natural projection of$C(g^{p})$is a self-dual code. We then discuss easy-to-handle criteria to decide if$C$is projective or not. As an application, we consider in the last part extremal self-dual codes of length 120, proving that their automorphism group does not contain elements of order 38 and 58.

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.

Let$C$be a binary extremal self-dual code of length 96. We prove that an automorphism of$C$of order 3 has 6 or no fixed points and an automorphism of order 5 has 6 fixed points. Moreover, if all automorphisms of order 3 are fixed point free then${\rm Aut}(C)$is solvable and its order divides$2^{5}3$or$2^{5}5$or${\rm Aut}(C)$is the alternating group${\rm A}_{5}$which is the only possible group of order 60. Furthermore,$\vert {\rm Aut}(C)\vert = 20$or$40$cannot occur.

We prove that the automorphism group of a binary self-dual doubly even [72, 36, 16] code has order 5, 7, 10, 14 ordwhereddivides 18 or 24, or it isA4×C3.

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 class of binary self-dual doubly even 2-quasi-cyclic transitive codes is asymptotically good. This improves a recent result of Bazzi and Mitter (IEEE Trans. Inf. Theory, vol. 52, pp. 3210-3219, 2006). The proof is based on the study of a particular class of codes invariant under dihedral groups using a blend of representation theory and probabilistic arguments. The methods are closely related to those used in Bazzi and Mitter. In order to complete the proof a number theoretical result of Hasse is needed.

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

There is the long-standing question whether the class of cyclic codes is asymptotically good. By an old result of Lin and Weldon, long Bose-Chaudhuri-Hocquenhem (BCH) codes are asymptotically bad. Berman proved that cyclic codes are asymptotically bad if only finitely many primes are involved in the lengths of the codes. We investigate further classes of cyclic codes which also turn out to be asymptotically bad. Based on reduction arguments we give some evidence that there are asymptotically good sequences of binary cyclic codes in which all lengths are prime numbers provided there is any asymptotically good sequence of binary cyclic codes.

Using representation theoretical methods we investigate self-dual group codes and their extensions in characteristic 2. We prove that the existence of a self-dual extended group code heavily depends on a particular structure of the group algebra KG which can be checked by an easy-to-handle criteria in elementary number theory. Surprisingly, in the binary case such a code is doubly even if the converse of Gleason's theorem holds true, i.e., the length of the code is divisible by 8. Furthermore, we give a short representation theoretical proof of an earlier result of Sloane and Thompson which states that a binary self-dual group code is never doubly even if the Sylow 2-subgroups of G are cyclic. It turns out that exactly in the case of a cyclic or Klein four group as Sylow 2-subgroup doubly even group codes do not exist.

We classify group algebras over Galois rings containing self-dual ideals; i.e., ideals C which satisfy C = C/sup /spl perp// with respect to the natural nondegenerate bilinear form given on group algebras.

Let C be an [n,k,d]/sub q/ linear code. The defect of C is the parameter s=s(C)=n-k+1-d. If k/spl ges/m+1/spl ges/2 then by the Griesmer bound d/spl les/(q/sup m/(q-1)/q/sub m/-1)(s+m). The author's interest is in those linear codes having the maximum minimum distance, i.e., d=(q/sup m/(q-1)/q/sup m/-1)(s+m). For m=1 we have d=q(s+1) and the codes are maximum minimum distance (MMD) codes in the sense of Faldum and Willems (see ibid., vol.44, p.1555-58, 1998). Thus we consider MMD codes in a more general sense. We refer to them simply as MMD codes. All MMD codes with m=1 are known up to formal equivalence. Note that two codes are formally equivalent if they have the same weight distribution. The author classifies up to formal equivalence the MMD codes with m/spl ges/2.

Let C be a linear [n,k,d]-code over GF(q) with k/spl ges/2. If s=n-k+1-d denotes the defect of C, then by the Griesmer bound, d/spl les/(s+1)q. Now, for obvious reasons, we are interested in codes of given defect s for which the minimum distance is maximal, i.e., d=(s+1)q. We classify up to formal equivalence all such linear codes over GF(q). Remember that two codes over GF(q) are formally equivalent if they have the same weight distribution. It turns out that for k/spl ges/3 such codes exist only in dimension 3 and 4 with the ternary extended Golay code, the ternary dual Golay code, and the binary even-weight code as exceptions. In dimension 4 they are related to ovoids in PG(3,q) except the binary extended Hamming code, and in dimension 3 to maximal arcs in PG(2,q).

Let C be an [n,k,d]-code over GP(q) with k/spl ges/2. Let s=def(C)=n+1-k-d denote the defect of C. The Griesmer bound implies that d/spl les/q(s+1). If d>qs and s/spl ges/2, then using a previous result of Faldum and Willems, k/spl les/q. Thus fixing s/spl ges/2 the extreme parameters for a code with def(C)=s are d=q(s+1); k=q, and n=k+d+s-1=(q+1)(s+2)-3. In this correspondence we characterize the codes with such parameters.