On the non-existence of perfect and nearly perfect codes

AbstractThe main result of the paper is the proof of the non-existence of a class of completely regular codes in certain distance-regular graphs. Corollaries of this result establish the non-existence of perfect and nearly perfect codes in the infinite families of distance-regular graphs J(2b + 1, b) and J(2b+2,b)

New Upper Bounds on Codes via Association Schemes and Linear Programming

Let A(n, d) denote the maximum number of codewords in a binary code of length n and minimum Hamming distance d. Upper and lower bounds on A(n, d) have been a subject for extensive research. In this paper we examine upper bounds on A(n, d) as a special case of bounds on the size of subsets in metric association scheme. We will first obtain general bounds on the size of such subsets, apply these bounds to the binary Hamming scheme, and use linear programming to further improve the bounds. We show that the sphere packing bound and the Johnson bound as well as other bounds are special cases of one of the bounds obtained from association schemes. Specific bounds on A(n, d) as well as on the sizes of constant weight codes are also discussed