Optimal Constant Weight Codes

A new class of binary constant weight codes is presented. We establish new lower bound and exact values on A(n, 2k, k + 1), in particular, A(30, 12, 7) = 9, A(48, 16, 9) = 11, A(51,16, 9) = 12, A(58, 18, 10) = 12. An ( ) w d n , , constant weight binary code is a code of length n , code distance d in which all code words have the same number of “ones” . The number of “ones” is w . We will denote the maximal possible size (a number of code words) of an ( ) w d n , , constant weight code by ( ) w d n A , , . The most important and interesting problem is finding the largest possible size ( ) w d n A , , of an ( ) w d n , , constant weight code (hereafter called optimal code). The results of code searching used to be put in tables of optimal codes. The first lower bound appeared in 1977 in the book of MacWilliams and Sloane ([1], pp.684-691). A table of binary constant weight codes of length 28 ≤ n with explicit constructions for most of the 600 codes was presented in the encyclopedic work of E E. Brouwer, J. B. Shearer, N. J .A .Sloane [2]. Today Neil J. A. Sloane presents his table of constant weight codes [3] online and performs continual updates. Let us consider an ( ) 1 , 2 , + k k n constant weight code with length n , code distance k 2 and weight of all 1 + k . Johnson’s upper bound (se, [2], p. 525) in this case is n k kn k k n A − + ≤ + 2 ) 1 ( ) 1 , 2 , ( (if denominator is positive) (1) Theorem. 2 ) 1 , 2 , 2 2 3 ( 2 + = + + + = k k k k k n A holds for all k. The research was support by The Royal Swedish academy of Sciences. Proof. The code is constructed from representation 2 2 3 2 + + k k element set } 2 2 3 ,..., 3 , 2 , 1 { 2 + + = k k M as union of 2 + k subsets i M with 1 + k elements Optimal Constant Weight Codes 913 in every subset and j i M M ∩ =1 for j i ≠ . We now give explicit construction of the code. For 2 = k and } 6 , 5 , 4 , 3 , 2 , 1 { = M we can find 4 subsets } 6 , 5 , 4 { }, 6 , 3 , 2 { }, 5 , 3 , 1 { }, 4 , 2 , 1 { 4 3 2 1 = = = = M M M M where all vectors of constant weight code ( ) 3 , 4 , 6 are: ) 1 , 1 , 1 , 0 , 0 , 0 ( ) 1 , 0 , 0 , 1 , 1 , 0 ( ) 0 , 1 , 0 , 1 , 0 , 1 ( ) 0 , 0 , 1 , 0 , 1 , 1 (