Complex Hadamard matrices of order 6: a four-parameter family

TL;DR

This paper introduces a new four-parameter family of complex Hadamard matrices of order 6, potentially completing the classification and advancing the understanding of the MUB-6 problem.

Contribution

It constructs a previously unknown four-parameter family of complex Hadamard matrices of order 6 using algebraic functions of roots of sextic polynomials.

Findings

New four-parameter family G of complex Hadamard matrices of order 6

Conjecture that G, K, and S exhaust all such matrices of order 6

Potential progress towards solving the MUB-6 problem

Abstract

In this paper we construct a new, previously unknown four-parameter family of complex Hadamard matrices of order 6, the entries of which are described by algebraic functions of roots of various sextic polynomials. We conjecture that the new, generic family G together with Karlsson's degenerate family K and Tao's spectral matrix S form an exhaustive list of complex Hadamard matrices of order 6. Such a complete characterization might finally lead to the solution of the famous MUB-6 problem.