2013/05/31 by Chunming Tang, Yanfeng Qi, Tang, Chunming +3 · 1 citation
Computer Science · Mathematics · #Cellular Automata and Applications #Coding theory and cryptography #Cryptographic Implementations and Security #FOS: Computer and information sciences #Information Theory (cs.IT) #cs.IT #math.IT
paper · pdf · doi:10.48550/arxiv.1305.7294
arxiv created 2013/05/31 · openalex publication_date 2013/05/31 · arxiv updated 2013/06/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Cyclic codes, as linear block error-correcting codes in coding theory, play a vital role and have wide applications. Ding in \citeD constructed a number of classes of cyclic codes from almost perfect nonlinear (APN) functions and planar functions over finite fields and presented ten open problems on cyclic codes from highly nonlinear functions. In this paper, we consider two open problems involving the inverse APN functions f(x)=xqm-2 and the Dobbertin APN function f(x)=x^24i+23i+22i+2i-1. From the calculation of linear spans and the minimal polynomials of two sequences generated by these two classes of APN functions, the dimensions of the corresponding cyclic codes are determined and lower bounds on the minimum weight of these cyclic codes are presented. Actually, we present a framework for the minimal polynomial and linear span of the sequence s∞ defined by st=Tr((1+αt)e), where α is a primitive element in GF(q). These techniques can also be applied into other open problems in \citeD.