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 subset of F-q(t) 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(a(1), a(2), ... , a(t)) = a(1), a(2), ... , a(t) is an element of F-pm}, where c(a(1), a(2), ... . a(t)) = (Tr-m(a(1)x(1) + a(2)x(2) + ... + a(t)x(t)))((x1, x2, ... , xt)) (is an element of D). In this paper, we will present the weight enumerators of the linear codes C-D in the following two cases: 1. D = {(x(1), x(2), ... , x(t)) is an element of F-q(t) \ {(0, 0, ... , 0)} : Tr-m(x(2)(1) + x(2)(2) + ... + x(2)(t)) = 0}; 2. D = {(x(1), x(2), ... , x(t)) is an element of F-q(t) : Tr-m(x(1)(2) + x(2)(2) + ... + x(t)(2)) = 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.