부산대학교 도서관

A theorem of McEliece on the $p$-divisibility of Hamming weights in cyclic codes over ${\BBF}_p$ is generalized to Abelian codes over ${{{\BBZ}/p^d{\BBZ}}}$. 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 ${{{\BBZ}/p^d{\BBZ}}}$. It also presents sharp results on $2$-divisibilities of Lee and Euclidean weights for Abelian codes over ${{{\BBZ}/4{\BBZ}}}$.