New code upper bounds for the folded n-cube

Abstract

Let Γ denote a distance-regular graph. The maximum size of codewords with minimum distance at least d is denoted by A(Γ, d). Let n denote the folded n-cube H(n, 2). We give an upper bound on A( n, d) based on block-diagonalizing the Terwilliger algebra of n and on semidefinite programming. The technique of this paper is an extension of the approach taken by A. Schrijver [8] on the study of A(H(n, 2), d).Observe that |x 1 △y 1 | = |x 2 △y 2 |, |x 1 △y 2 | = |x 2 △y 1 |, and |x 1 △y 1 | + |x 1 △y 2 | = n. Then it follows that ∂(x, y) = min{|x 1 △y 1 |,|x 1 △y 2 |} and 0 ≤ ∂(x, y) ≤ ⌊ n 2 ⌋, where ⌊a⌋ denotes the maximal integer less than or equal to a. It is well-known that n is a bipartite (an almost-bipartite) distance-regular graph with diameter ⌊ n 2 ⌋ for even n (odd n). The paper is organized as follows. In Section 2, we recall some definitions and facts concerning the distance-regular graph and its Terwilliger algebra. In Section 3, we give a basis of the Terwilliger algebra of n by considering the action of automorphism group of n on X × X × X. In Section 4, we study a block-diagonalization of the Terwilliger algebra via the obtained basis. In Section 5, we estimate an upper bound on A( n , d) by semidefinite programming involving the block-diagonalization of the Terwilliger algebra. Moreover, we offer several concrete upper bounds on A( n , d) for 8 ≤ n ≤ 13.

The Terwilliger algebra of the halved n-cube from the viewpoint of its automorphism group action

“…(i) For i being even and k being odd, we use induction on k (k ≥ 1) to show (19). By (7), it is easy to see (19) holds for k = 1. We now assume that (19) holds for k − 2 (k ≥ 3), that is,…”

“…x ∩ y = ∅, and 0 otherwise. (iii) Take transpose of matrices at both sides of (7). (iv) Similar to the proof of (ii).…”

“…In general, the above two algebras do not coincide. However, it was proved that the two algebras are equal for some distance-regular graphs: the Hamming graphs ( [6,10]), the Johnson graphs ( [11]), the folded n-cubes ( [7]), the halved n-cubes ( [8]) and the halved folded 2n-cubes ( [4]). An interesting and important problem is to find when the two algebras coincide.…”