2020/12/11 by Dabin Zheng, Xiaoqiang Wang, Zheng, Dabin +5
Computer Science · Social Sciences · #Advanced Data Storage Technologies #Coding theory and cryptography #FOS: Computer and information sciences #Information Theory (cs.IT) #Islamic Finance and Communication
paper · pdf · doi:10.48550/arxiv.2012.06105
openalex publication_date 2020/12/11 · openalex created_date 2022/09/13 · openalex updated_date 2026/07/28
Let \mathbbFpm be a finite field with pm elements, where p is an odd prime and m is a positive integer. Recently, \citeHengar and \citeWang2020 determined the weight distributions of subfield codes with the form Cf=\(( \rm Tr1m(a f(x)+bx)+c)_x ∈ \mathbbFpm, \rm Tr1m(a)) : a,b ∈ \mathbbFpm, c ∈ \mathbbFp\ for f(x)=x2 and f(x)=xpk+1, respectively, where k is a nonnegative integer. In this paper, we further investigate the subfield code Cf for f(x) being a known perfect nonlinear function over \mathbbFpm and generalize some results in \citeHengar,Wang2020. The weight distributions of the constructed codes are determined by applying the theory of quadratic forms and the properties of perfect nonlinear functions over finite fields. In addition, the parameters of the duals of these codes are also determined. Several examples show that some of our codes and their duals have the best known parameters with respect to the code tables in \citeMGrassl. The duals of some proposed codes are optimal with respect to the Sphere Packing bound if p≥ 5.