Hadamard bent sequences and spherical designs

Official URL: https://dx.doi.org/10.1007/s10801-025-01463-x

In (Solé et al, ISIT 2021), a new notion of bent sequences appeared, where the Sylvester matrix of the Walsh–Hadamard transform is replaced by an arbitrary Hadamard matrix. We introduce an even more general notion of a bent sequence, defined in terms of the following data: a Hadamard matrix of order n defined over the complex qth roots of unity (a so-called Butson matrix), an algebraic integer in the cyclotomic field of order q, and a Galois automorphism of that field. This new generalization facilitates the existence of self-dual bent sequences in cases where the restricted definition fails. In particular, we construct self-dual bent sequences for various q≤60 and lengths n≤21. Computational construction methods comprise the resolution of polynomial systems by Gröbner bases and eigenspace computations. Infinite families of self-dual bent sequences can be constructed from regular Hadamard matrices, Bush-type Hadamard matrices, and generalized Boolean bent functions. As an application of Hadamard bent sequences, we derive a lower bound on the covering radius of the Zq-code attached to a Hadamard matrix, for the so-called Chinese Euclidean metric. To every Hadamard matrix we attach a spherical code, which is optimal as a packing code when q=4. This code is a strength 2 spherical design under some mild conditions. Based on the covering properties of spherical designs, we bound above the covering radius of that spherical code for the standard Euclidean metric. This upper bound, in turn, yields an upper bound on the covering radius of the Zq-code for the Chinese Euclidean metric which is not far from the lower bound.