OPTIMAL LINEAR CODES OVER ℤ_m

Abstract

We examine the main linear coding theory problem and study the structure of optimal linear codes over the ring <TEX>${\mathbb{Z}}_m$</TEX>. We derive bounds on the maximum Hamming weight of these codes. We give bounds on the best linear codes over <TEX>${\mathbb{Z}}_8$</TEX> and <TEX>${\mathbb{Z}}_9$</TEX> of lengths up to 6. We determine the minimum distances of optimal linear codes over <TEX>${\mathbb{Z}}_4$</TEX> for lengths up to 7. Some examples of optimal codes are given.