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

^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.