2023/02/10 by Martin Scotti, Scotti, Martin · 2 citations
Computer Science · Engineering · Mathematics · #Coding theory and cryptography #Combinatorics (math.CO) #FOS: Computer and information sciences #FOS: Mathematics #Finite Group Theory Research #Information Theory (cs.IT) #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.2302.05350
openalex publication_date 2023/02/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In recent years, many connections have been made between minimal codes, a classical object in coding theory, and other remarkable structures in finite geometry and combinatorics. One of the main problems related to minimal codes is to give lower and upper bounds on the length m(k,q) of the shortest minimal codes of a given dimension k over the finite field \mathbbFq. It has been recently proved that m(k, q) ≥ (q+1)(k-1). In this note, we prove that \liminfk → ∞ (m(k, q))/(k) ≥ (q+ ε(q) ), where ε is an increasing function such that 1.52 <ε(2)≤ ε(q) ≤ √(2) + (1)/(2). Hence, the previously known lower bound is not tight for large enough k. We then focus on the binary case and prove some structural results on minimal codes of length 3(k-1). As a byproduct, we are able to show that, if k = 5 \pmod 8 and for other small values of k, the bound is not tight.