A class of subfield codes and their duals
Huang Junsong, Tongjiang Yan · Advances in Mathematics of Communications · 2025
Recently, Xiang and Yin studied two families of subfield codes of linear codes over $ \mathbb{F}_{2^m} $, and determined the weight distribution of the linear code$ \mathcal{C}^{(2)} = \left\{\left(\left(\operatorname{Tr}_{q/2}(ax^2+by^2+wy)+c\right)_{x,y\in\mathbb{F}_q},\operatorname{Tr}_{q/2}(a)\right):a,b,w\in\mathbb{F}_q,c\in\mathbb{F}_2\right\}, $where $ q = 2^m $. Motivated by this paper and other former contributions, we study the subfield codes of linear codes over $ \mathbb{F}_{p^m} $ and obtain the weight distribution of$ \mathcal{C}^{(p)} = \left\{\left(\operatorname{Tr}_{p^m/p}(ax^{p^k+1}+bx+uy)+w\right)_{x,y \in \mathbb{F}_{p^m}}: a, b,u \in \mathbb{F}_{p^m}, w \in \mathbb{F}_p\right\}, $which is a self-orthogonal $ p $-divisible linear code. We also investigate the dual codes of $ \mathcal{C}^{(p)} $, whose parameters are obtained by the weight distribution of $ \mathcal{C}^{(p)} $, and they are almost optimal codes.