A lemma on polynomials modulo pmpm and applications to coding theory

An integer-valued function f(x)f(x) on the integers that is periodic of period pepe, p   prime, can be matched, modulo pmpm, by a polynomial function w(x)w(x); we show that w(x)w(x) may be taken to have degree at most (m(p-1)+1)pe-1-1(m(p-1)+1)pe-1-1. Applications include a short proof of the theorem of McEliece on the divisibility of weights of codewords in p-ary cyclic codes by powers of p, an elementary proof of the Ax–Katz theorem on solutions of congruences modulo p, and results on the numbers of codewords in p  -ary linear codes with weights in a given congruence class modulo pepe.