A lemma on polynomials modulo image and applications to coding theory Original Research Article

An integer-valued function image on the integers that is periodic of period image, p prime, can be matched, modulo image, by a polynomial function image; we show that image may be taken to have degree at most image. 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 image.