The weight distributions of cosets are not known for the class of binary 2-error-correcting BCH codes of length n=2/sup m/-1, m even (the nonuniformly packed case). By using the graph theoretical concepts of the combinatorial matrix of a code and an r-partition design introduced in previous works, the authors obtain these distributions of cosets in the three particular cases of length 15, 63, and 255. It is observed that in these three cases the number of distinct weight distributions is a constant equal to 8.>

Letf(x)\in {\bf F}_{q}[X]be a polynomial with simple roots to be factored. The so-called Berlekamp subalgebraBspanned over{\bf F}_{q}by the idempotents ofA={\bf F}_{q}[X]/(f(X))is considered. An exponential technique introduced earlier is based upon taking elements from B at random and enables us to obtain idempotents and, from that, the factors of f(X). This algorithm is speeded up in three ways. The concept of a separating subset of B is introduced and the McEliece operator mapping A onto B is used to construct a small separating set. {\em Factoring} subsets of{\bf F}_{q}were defined and investigated previously. The algorithm and these subsets are used together with a process introduced by F. J. McWilliams for the rapid construction of primitive idempotents. Finally, an algorithm is introduced for constructing irreducible polynomials of{\bf F}_{q}[X]of degreed, for large values ofd, in which the most expensive operation is the Euclidian algorithm applied to two polynomials of degree2d.