Several classes of linear codes with AMDS duals and their subfield codes

Abstract

Subfield codes of linear codes over finite fields have recently attracted great attention due to their wide applications in secret sharing, authentication codes and association schemes. In this paper, we first present a construction of 3-dimensional linear codes \(\varvec{C}_{\varvec{f}}\) over finite field \({\mathbb {F}_{\varvec{2}}}^{\varvec{m}}\) parameterized by any Boolean function \(\varvec{f}\). Then we determine explicitly the weight distributions of \(\varvec{C}_{\varvec{f}}\), the punctured code \(\widetilde{\varvec{C}}_{\varvec{f}}\), as well as the corresponding subfield codes over \(\mathbb {F}_{\varvec{2}}\) for several classes of Boolean functions \(\varvec{f}\). In particular, we determine the weight distributions of subfield codes derived from \(\varvec{r}\)-plateaued functions. Moreover, the parameters of their dual codes are investigated, which contain length-optimal and dimension-optimal AMDS codes with respect to the sphere packing bound. We emphasize that the new codes are projective and contain binary self-complementary codes. As applications, some of the projective codes we present can be employed to construct \(\varvec{s}\)-sum sets for any odd integer \(\varvec{s}>\varvec{1}\).