We study the coset weight distributions of the 3-error-correcting binary narrowsense BCH-codes and of their extensions, whose lengths are, respectively, 2m − 1 and 2m, m odd. We prove that all weight distributions are known as soon as those of the cosets of minimum weight 4 of the extended code are known. We point out that properties of the cosets which are orphans yield interesting properties ...

Let H,K be subgroups of G. We investigate the intersection properties of left and right cosets of these subgroups. If H and K are subgroups of G, then G can be partitioned as the disjoint union of all left cosets of H, as well as the disjoint union of all right cosets of K. But how do these two partitions of G intersect each other? Definition 1. Let G be a group, and H a subgroup of G. A left t...

We study a family of particular cosets of the first-order Reed–Muller code R(1;m): those generated by special codewords, the idempotents. Thus we obtain new maximal weight distributions of cosets of R(1; 7) and 84 distinct almost maximal weight distributions of cosets of R(1; 9), that is, with minimum weight 240. This leads to crypotographic applications in the context of stream ciphers.

We study incidence properties among cosets of finite loops, with emphasis on well-structured varieties such as antiautomorphic loops and Bol loops. While cosets in groups are either disjoint or identical, we find that the incidence structure in general loops can be much richer. Every symmetric design, for example, can be realized as a canonical collection of cosets of a finite loop. We show tha...

We show how to gauge the set of raising and lowering generators of an arbitrary Lie algebra. We consider SU(N) as an example. The nilpotency of the BRST charge requires constraints on the ghosts associated to the raising and lowering generators. To remove these constraints we add further ghosts and we need a second BRST charge to obtain nontrivial cohomology. The second BRST operator yields a g...