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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… (More)
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… (More)
In  we studied Euclidean self-dual extended group codes over finite fields K of characteristic 2. The groups G involved in the investigations had to be of odd order which means that the corresponding group algebras were semisimple. The situation becomes more subtle if we drop the assumption charK = 2 since in that case KG is no longer semisimple.… (More)
We show that Brin’s generalisations 2V and 3V of the Thompson-Higman group V are of type FP∞. Our methods also give a new proof that both groups are finitely presented.
The weight of a code is the number of coordinate positions where no codeword is zero. The rth minimum weight dr is the least weight of an r-dimensional subcode. Wei and Yang conjectured a formula for the minimum weights of some product codes. The conjecture is proved in two di erent ways, each with interesting side-results.
General arguments of Baumslag and Bieri guarantee that any torsion-free finitely generated metabelian group of finite Prüfer rank can be embedded in a metabelian constructible group. Here, we consider the metric behavior of a rich class of examples and analyze the distortions of specific embeddings.
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 (<i>IEEE Trans. Inf. Theory</i>, 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… (More)