Sharp $p$ -Divisibility of Weights in Abelian Codes Over ${BBZ}/p^d{BBZ}$

A theorem of McEliece on the p-divisibility of Hamming weights in cyclic codes over F_p is generalized to Abelian codes over Zopf/p ^dZopf. 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 Zopf/p ^dZopf. It also presents sharp results on 2-divisibilities of Lee and Euclidean weights for Abelian codes over Zopf/4Zopf.