2025/06/09 by Haihua Deng, Deng, Haihua, Hexiang Huang +3
Computer Science · #11A07 #11T71 #94B05 #Coding theory and cryptography #Combinatorics (math.CO) #Cooperative Communication and Network Coding #Error Correcting Code Techniques #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.2506.07846
openalex publication_date 2025/06/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we consider Griesmer codes, namely those linear codes meeting the Griesmer bound. Let C be an [n,k,d]q Griesmer code with q=pf, where p is a prime and f≥1 is an integer. In 1998, Ward proved that for q=p, if pe|d, then pe|wt(c) for all c∈ C. In this paper, we show that if qe|d, then C has a basis consisting of k codewords such that the first min\e+1,k\ of them span a Griesmer subcode with constant weight d and any k-1 of them span a [gq(k-1,d),k-1,d]q Griesmer subcode. Using the p-adic algebraic method together with this basis, we prove that if qe|d, then pe|wt(c) for all c∈ C. Based on this fact, using the geometric approach with the aforementioned basis, we show that if pe|d, then Δ|\rm wt(c) for all c∈ C, where Δ=\lceil pe-(f-1)(q-2)\rceil.