2019/10/12 by Wang, Xiaoqiang, Zheng, Dabin, Zhang, Yan
#FOS: Computer and information sciences #Information Theory (cs.IT)
paper · doi:10.48550/arxiv.1910.05461
Let \mathbbFpm be the finite field of order pm, where p is an odd prime and m is a positive integer. In this paper, we investigate a class of subfield codes of linear codes and obtain the weight distribution of \beginsplit Ck=\(( \rm Tr1m(axpk+1+bx)+c)_x ∈ \mathbbFpm, \rm Tr1m(a)) : a,b ∈ \mathbbFpm, c ∈ \mathbbFp\, \endsplit where k is a nonnegative integer. Our results generalize the results of the subfield codes of the conic codes in \citeHengar. Among other results, we study the punctured code of Ck, which is defined as \mathcalCk=\( \rm Tr1m(a x^pk+1+bx)+c)_x ∈ \mathbbFpm : a,b ∈ \mathbbFpm, c ∈ \mathbbFp\. The parameters of these linear codes are new in some cases. Some of the presented codes are optimal or almost optimal. Moreover, let v2(⋅) denote the 2-adic order function and v2(0)=∞, the duals of Ck and \mathcalCk are optimal with respect to the Sphere Packing bound if p>3, and the dual of \mathcalCk is an optimal ternary linear code for the case v2(m)≤ v2(k) if p=3 and m>1.