2023/12/01 by Chubenko, Vladimir, Kurz, Sascha · 1 citation
Computer Science · Engineering · #94B05 (51E23) #Coding theory and cryptography #Combinatorics (math.CO) #Cooperative Communication and Network Coding #FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT) #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.2312.00885
openalex publication_date 2023/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Minimal codes are linear codes where all non-zero codewords are minimal, i.e., whose support is not properly contained in the support of another codeword. The minimum possible length of such a k-dimensional linear code over \mathbbFq is denoted by m(k,q). Here we determine m(7,2), m(8,2), and m(9,2), as well as full classifications of all codes attaining m(k,2) for k≤ 7 and those attaining m(9,2). We give improved upper bounds for m(k,2) for all 10≤ k≤ 17. It turns out that in many cases the attaining extremal codes have the property that the weights of all codewords are divisible by some constant Δ>1. So, here we study the minimum lengths of minimal codes where we additionally assume that the weights of the codewords are divisible by Δ. As a byproduct we also give a few binary linear codes improving the best known lower bound for the minimum distance.