2024/02/20 by Ivan Landjev, Landjev, Ivan, Konstantin Vorob’ev +1
Computer Science · Engineering · #05A05 #05A20 #94B25 #94B65 #Coding theory and cryptography #Combinatorics (math.CO) #Cooperative Communication and Network Coding #FOS: Mathematics #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.2402.13420
openalex publication_date 2024/02/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider the problem of finding A2(n,\d1,d2\) defined as the maximal size of a binary (non-linear) code of length n with two distances d1 and d2. Binary codes with distances d and d+2 of size ∼(n2)/((d)/(2)((d)/(2)+1)) can be obtained from 2-packings of an n-element set by blocks of cardinality (d)/(2)+1. This value is far from the upper bound A2(n,\d1,d2\)≤1+n\choose2 proved recently by Barg et al. In this paper we prove that for every fixed d (d even) there exists an integer N(d) such that for every n≥ N(d) it holds A2(n,\d,d+2\)=D(n,(d)/(2)+1,2), or, in other words, optimal codes are isomorphic to constant weight codes. We prove also estimates on N(d) for d=4 and d=6.