Some two-weight and three-weight linear codes

Let F_q be the finite field with q = p^m elements, where p is an odd prime and m is a positive integer. For a positive integer t, let D ⊂ F^t _q and let Tr_m be the trace function from F_q onto F_p. We define a p-ary linear code C_D by C_D = {c(a₁, a₂, …, a_t): a₁, a₂, …, a_t ∈ F_p ^m}, where c(a₁, a₂, …, a_t) = (Tr_m(a₁x₁ + a₂x₂ + · · · + a_tx_t))_{(x1,x2,…,xt) ∈D}. In this paper, we will present the weight enumerators of the linear codes C_D in the following two cases: 1. D = {(x₁, x₂, …, x_t) ∈ F^t _q \ {(0, 0, …, 0)}: Tr_m(x² ₁ + x² ₂ + · · · + x² _t) = 0}; 2. D = {(x₁, x₂, …, x_t) ∈ F^t _q: Tr_m(x² ₁ + x² ₂ + · · · + x² _t) = 1}. It is shown that C_D is a two-weight code if tm is even and three-weight code if tm is odd in both cases. The weight enumerators of C_D in the first case generalize the results in [17] and [18]. The complete weight enumerators of C_D are also investigated.