A lemma on polynomials modulo p^m and applications to coding theory

An integer-valued function f(x) on the integers that is periodic of period p^e, p prime, can be matched, modulo p^m, by a polynomial function w(x); we show that w(x) may be taken to have degree at most (m(p-1)+1)p^(e-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 p^e