Wolfgang Willems

ON THE NUMBER OF COMPONENTS OF A GRAPH RELATED TO CHARACTER DEGREES

We connect two nonlinear irreducible character of a finite group G if their degrees have a common prime divisor. In this paper we show that the corresponding graph has at most three connected components.

A note on self-dual group codes

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.

Codes of Small Defect

The parameters of a linear code C over GF(q) are given by [n,k,d], where n denotes the length, k the dimension and d the minimum distance of C. The code C is called MDS, or maximum distance separable, if the minimum distance d meets the Singleton bound, i.e. d = n-k+1 Unfortunately, the parameters of an MDS code are severely limited by the size of the field. Thus we look for codes which have minimum distance close to the Singleton bound. Of particular interest is the class of almost MDS codes, i.e. codes for which d=n-k. We will present a condition on the minimum distance of a code to guarantee that the orthogonal code is an almost MDS code. This extends a result of Dodunekov and Landgev Dodunekov. Evaluation of the MacWilliams identities leads to a closed formula for the weight distribution which turns out to be completely determined for almost MDS codes up to one parameter. As a consequence we obtain surprising combinatorial relations in such codes. This leads, among other things, to an answer to a question of Assmus and Mattson 5 on the existence of self-dual [2d,d,d]-codes which have no code words of weight d+1. Actually there are more codes than Assmus and Mattson expected, but the examples which we know are related to the expected ones.

Projective two-weight codes with small parameters and their corresponding graphs

Projective two-weight codes with relatively small parameters are enumerated up to equivalence. Some properties of codes and related strongly-regular graphs are presented.

On the Crosscorrelation of Sequences with the Decimation Factor d = (p n +1)/(p+1) - (p n -1)/2.

Abstract. Let p be a prime with p≡3 (mod 4), let n be an odd natural number and let . Consider the crosscorrelation funtion Cd(t)=∑i=1pn−1ζai−adi−t where ζ≠1 is a complex p-th root of unity and (ai) is a maximal linear shift register sequence. In 7 the bound has been computed for p = 3. In this note we generalize this to for p≥ 3. Furthermore we giv an upper bound for the probability of the crosscorrelation function achieving the maximum absolute value.

Is the class of cyclic codes asymptotically good

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.

Self-dual codes and modules for finite groups in characteristic two

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 Gleasons 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.

Automorphisms of Extremal Self-Dual Codes

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

A characterization of MMD codes

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).

On the Automorphism Group of a Binary Self-Dual Doubly Even

We prove that the automorphism group of a binary self-dual doubly even [72, 36, 16] code has order 5, 7, 10, 14 or <i>d</i> where <i>d</i> divides 18 or 24, or it is <i>A</i>₄ × <i>C</i>₃.