Sharp p-Divisibility of Weights in Abelian Codes Over Z/pdZ.

A theorem of McEliece on the p-divisibility of Hamming weights in cyclic codes over Fp is generalized to Abelian codes over Z/pd Z. This work improves upon results of Helleseth—Kumar—Moreno—Shanbhag, Calderbank—Li—Poonen, Wilson, and Katz. These previous attempts are not sharp in general, i.e., do not report the full extent of the p-divisibility except in special cases, nor do they give accounts of the precise circumstances under which they do provide best possible results. This paper provides sharp results on p-divisibilities of Hamming weights and counts of any particular symbol for an arbitrary Abelian code over Z/pd Z. It also presents sharp results on 2-divisibilities of Lee and Euclidean weights for Abelian codes over Z/4Z. [ABSTRACT FROM AUTHOR]