A construction of several classes of two-weight and three-weight linear codes

Linear codes constructed from defining sets have been extensively studied and may have a few nonzero weights if the defining sets are well chosen. Let F_q be a finite field with q= p^m elements, where p is a prime and m is a positive integer. Motivated by Ding and Ding’s recent work (IEEE Trans Inf Theory 61(11):5835–5842, 2015), we construct p-ary linear codes C_D by CD={c(a,b)=(Trm(ax+by))(x,y)∈D:a,b∈Fq},where D⊂Fq2 and Tr _m is the trace function from F_q onto F_p. In this paper, we will employ exponential sums to investigate the weight enumerators of the linear codes C_D, where D={(x,y)∈Fq2\{(0,0)}:Trm(xN1+yN2)=0} for two positive integers N₁ and N₂. Several classes of two-weight and three-weight linear codes and their explicit weight enumerators are presented if N1,N2∈{1,2,pm2+1}. By deleting some coordinates, more punctured two-weight and three-weight linear codes C_D¯ which include some optimal codes are derived from C_D.