Quasi-unbiased Hadamard matrices and weakly unbiased Hadamard matrices: A coding-theoretic approach

Abstract

This paper is concerned with quasi-unbiased Hadamard matrices and weakly unbiased Hadamard matrices, which are generalizations of unbiased Hadamard matrices, equivalently unbiased bases. These matrices are studied from the viewpoint of coding theory. As a consequence of a coding-theoretic approach, we provide upper bounds on the number of mutually quasi-unbiased Hadamard matrices. We give classifications of a certain class of self-complementary codes for modest lengths. These codes give quasi-unbiased Hadamard matrices and weakly unbiased Hadamard matrices. Some modification of the notion of weakly unbiased Hadamard matrices is also provided.

Semidefinite programming bounds for complex spherical codes

“…In Section 6, we deal with real spherical 3-distance sets related to Hadamard matrices and association schemes. We improve the known upper bounds obtained by the linear programming in [1] and reprove a result in association schemes.…”

“…(b) and A the adjacency matrix of the tournament attached to X. Then the Gram matrix of X is the same as the Gram matrix of the image of the minimal representation of the tournament [12,Theorem 4.8 (ii)], and the Gram matrix of X is given by (1) [12,Theorem 3.1 (3)]. After the suitable permuting the rows and the columns simultaneously the matrix (…”

Unbiased orthogonal designs

“…Using this equality, it is easily shown that for weighing matrices W 1 , W 2 of order n and weight k, W 1 , W 2 are quasi-unbiased for parameters (n, k, l, a) if and only if (1/ √ a)W 1 W ⊤ 2 is a (0, 1, −1)-matrix. The case for the parameters (n, n, l, a) was studied in [2] from the viewpoint of coding theory.…”

“…It is known that if there exist Hadamard matrices of order 4m, 4n, then there exists a Hadamard matrix of order 8mn [1]. This construction was used to construct quasi-unbiased Hadamard matrices in [2]. We use this idea to orthogonal designs in order to obtain unbiased orthogonal designs.…”

“…Since there exist Williamson type matrices of order 3, Proposition 4.11 provides four mutually quasi-unbiased Hadamard matrices for the parameters (48, 48, 16, 144) are obtained. These parametrs were missing from Table 1 in [2].…”