Two-weight codes: Upper bounds and new optimal constructions

• Published in: Discrete mathematics 2019-11, Vol.342 (11), p.3098-3113
• Main Authors: Lan, Liantao, Chang, Yanxun
• Format: Article
• Language: English
• Subjects: Group divisible packing; Multi-weight; Optimal; Two-weight code
• ISSN: 0012-365X; 1872-681X
• DOI: 10.1016/j.disc.2019.06.019

In this paper, we consider q-ary codes of length n and minimum Hamming distance d, which have two weights w1 and w2. These codes are denoted by (n,d,{w1,w2})q codes. Let Aq(n,d,{w1,w2}) denote the largest possible number of codewords in an (n,d,{w1,w2})q code and we simply write A(n,d,{w1,w2}) for A2(n,d,{w1,w2}). Some upper bounds on Aq(n,d,{w1,w2}) are given. The equivalences between binarytwo-weight codes and special combinatorial configurations with certain properties are established and then new upper bounds on A(n,d,{w1,w2}) are derived. For w1,w2∈{2,3,4}, optimal constructions of (n,d,{w1,w2})2 codes are presented. The exact value of A(n,d,{2,3}) is completely determined for all n and d. We determine the exact value of A(n,d,{2,4}) for any positive integer n≡2,4(mod6) and d∈{3,4}. The exact value of A(n,d,{3,4}) is determined for any integer n and d∈{6,7}, or d=5 and n≡0,2,3,11(mod12).