Abstract-
I. DESCRIPTION OF THE METHODWe present a new upper bound on A(n, d), the maximum size of a binary code of word length n and minimum distance at least d. The bound is based on block-diagonalising the (noncommutative) Terwilliger algebra of the Hamming cube and on semidefinite programming. The bound refines the Delsarte bound [4], which is based on diagonalising the (commutative) Bose-Mesner algebra of the Hamming cube and on linear programming. We describe the approach in this section, and go over to the details in Section II.Taking a tensor product of the algebra, this approach also yields a new upper bound on A (n, d, w), the maximum size of a binary code of word length n, minimum distance at least d, and constant weight w. This bound strengthens the Delsarte bound for constant-weight codes. We describe this method in Section III.Fix a nonnegative integer n, and let P be the collection of all subsets of {1, . . . , n}. We identify code words in {0, 1} n with their support. So a code C is a subset of P. The Hamming distance of X, Y ∈ P is equal to |X Y |. The minimum distance of a code C is the minimum Hamming distance of distinct elements of C. For finite sets U and V , a U ×V matrix is a matrix whose rows and columns are indexed by U and V , respectively.For background on coding theory and association schemes we refer to MacWilliams and Sloane [9]. However, most of this paper is self-contained. While we will mention below a theorem on the existence of a block-diagonalisation of a C * -algebra, we prove this theorem for the algebras concerned by displaying an explicit block-diagonalisation. CWI
A. The Terwilliger algebraWe first describe the Terwilliger algebra of the Hamming cube, in a form convenient for our purposes. For background we refer to our notes in Subsection I-C.For nonnegative integers i, j, t, let M t i,j be the P ×P matrix withLet A n be the set of matrices (2) n i,j,t=0It is easy to check that A n is a C * -algebra: it is closed under addition, scalar and matrix multiplication, and taking the adjoint. since it is the number of triples (i, j, t) with M t i,j = 0, which is equal to the number of triples (a, b, t) with a + b + t ≤ n.