id: 06110838
dt: j
an: 06110838
au: Katz, Daniel J.
ti: On theorems of Delsarte-McEliece and Chevalley-warning-Ax-Katz.
so: Des. Codes Cryptography 65, No. 3, 291-324 (2012).
py: 2012
pu: Springer, Norwell, MA
la: EN
cc: 
ut: cyclic codes; abelian codes; algebraic sets; Delsarte-McEliece; Ax-Katz
ci: 
li: doi:10.1007/s10623-012-9645-y
ab: Summary: We present a theorem that generalizes the result of Delsarte and
    McEliece on the $p$-divisibilities of weights in abelian codes. Our
    result generalizes the Delsarte-McEliece theorem in the same sense that
    the theorem of N. M. Katz generalizes the theorem of Ax on the
    $p$-divisibilities of cardinalities of affine algebraic sets over
    finite fields. As the Delsarte-McEliece theorem implies the theorem of
    Ax, so our generalization implies that of N. M. Katz. The generalized
    theorem gives the $p$-divisibility of the $t$-wise Hamming weights of
    $t$-tuples of codewords $(c^{(1)},\dots,c^{(t)})$ as these words range
    over a product of abelian codes, where the $t$-wise Hamming weight is
    defined as the number of positions $i$ in which the codewords do not
    simultaneously vanish, i.e., for which
    $(c^{(1)}_i\ldots,c^{(t)}_i)\not=(0,\dots,0)$. We also present a
    version of the theorem that, for any list of $t$ symbols
    $s_1,\dots,s_t$, gives $p$-adic estimates of the number of positions
    $i$ such that $(c^{(1)}_i,\dots,c^{(t)}_i)=(s_1,\dots,s_t)$ as these
    words range over a product of abelian codes.
rv: