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

Daniel J Katz

doi:10.1109/tit.2008.2006434

Abstract

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 d Zopf. 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 d Zopf. It also presents sharp results on 2 -divisibilities of Lee and Euclidean weights for Abelian codes over Zopf /4 Zopf.

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