BOUNDS ON UNIQUE-NEIGHBOR CODES

Recall that a binary linear code of length n is a linear subspace (formula presenetd). Heretheparitycheckmatrix Aisabinarym×nmatrixofrankm. WesaythatC has rate R = 1−m/n. Itsdistance, denoted δn is the smallest Hamming weight of a non-zero vector in C. The rate vs. distance problem for binary linear codes is a fundamental open problem in coding theory, and a fascinating question in discrete mathematics. It concerns the function R_L(δ), the largest possible rate R for given (formula presenetd) and arbitrarily large length n. Here we investigate a variation of this fundamental question that we describe next. Clearly, C has distance δn, if and only if for every 0 n′ δ_n, every m × n′ sub matrix of A has a row of odd weight. Motivated by several problems from coding theory, we say that A has the unique-neighbor property with parameter δ_n, if every such subma trix has a row of weight 1. Let R_U(δ) be the largest possible asymptotic rate of linear codes with a parity check matrix that has this stronger property. Clearly, R_U(•),R_L(•) are non-increasing functions, and (formual presented) for all δ. Also, R_U(0) = R_L(0) = 1, and R_U(1) = R_L(1) = 0, so let (forula presented) 1bethesmallest values of δ at which R_U resp. RL vanish. It is well known that δ_L = 1/2 and we conjecture that δ_U is strictly smaller than 1 2, i.e., the rate of linear codes with the unique-neighbor property is more strictly bounded. While the conjecture remains open, we prove here several results supporting it. The reader is not assumed to have any specific background in coding theory, but we occasionally point out some relevant facts from that area.