Several new classes of linear codes with few weights

Abstract

Let $\phantom {\dot {i}\!}\mathbb {F}_{q}$ be a finite field of order q, where q = p^s is a power of a prime number p. Let m and m₁ be two positive integers such that m₁ divides m. For any positive divisor e of q − 1, we construct an infinite family of codes with dimension m + m₁ and few weights over $\phantom {\dot {i}\!}\mathbb {F}_{q}$. Using Gauss sum, their weight distributions are provided. When gcd(e, m) = 1, we obtain a subclass of optimal codes which attain the Griesmer bound. Moreover, when gcd(e, m) = 2 or 3 we construct new infinite families of codes with at most four weights.