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 \mathbb {F}}_q$$Fq be a finite field with $q=p^m$$q=pm 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 \mathcal {C}}_D$$CD by \begin{aligned} \mathcal {C}}_D=\mathbf {c}(a,b)\big \text {Tr}_m(ax+by)\big )_{(x,y)\in D} a, b \in \mathbb {F}}_q\ \end{aligned}$CD={c(a,b)(Trm(ax+by)(x,y)D:a,bFq},where $D \subset \mathbb {F}}_q^2$$DFq2 and \text {Tr}_m$$Trm is the trace function from \mathbb {F}}_q$$Fq onto \mathbb {F}}_p$$Fp. In this paper, we will employ exponential sums to investigate the weight enumerators of the linear codes \mathcal {C}}_D$$CD, where $D=(x, y) \in \mathbb {F}}_q^2 \setminus (0,0) \text {Tr}_m(x^{N_1}+y^{N_2}=0\$D=(x,y)Fq2\(0,0):Trm(xN1+yN2)=0} for two positive integers $N_1$$N1 and $N_2$$N2. Several classes of two weight and three weight linear codes and their explicit weight enumerators are presented if $N_1, N_2 \in {1, 2, p^\frac{m}{2}+1\$N1,N2{1,2,pm2+1} By deleting some coordinates, more punctured two weight and three weight linear codes \mathcal {C}}_{\overline{D}$CD which include some optimal codes are derived from \mathcal {C}}_D$$CD.
Applicable Algebra in Engineering, Communication and Computing