[Paper Review] Threeparameter complex Hadamard matrices of order 6

Bengt R. Karlsson|arXiv (Cornell University)|Mar 22, 2010

graph theory and CDMA systems4 citations

TL;DR

This paper presents a three-parameter family of complex Hadamard matrices of order 6, significantly extending known one- and two-parameter families by unifying them as subfamilies. The construction relies on $H_2$-reducibility and unitarity constraints, yielding a complete characterization of $H_2$-reducible matrices through Möbius transformations and unitary matrices, with implications for higher-dimensional constructions.

ABSTRACT

A three-parameter family of complex Hadamard matrices of order 6 is presented. It significantly extends the set of closed form complex Hadamard matrices of this order, and in particular contains all previously described one- and two-parameter families as subfamilies.

Motivation & Objective

• To provide a complete parametrization of $H_2$-reducible complex Hadamard matrices of order 6.
• To unify previously known one- and two-parameter families as subfamilies within a single three-parameter family.
• To extend the classification of complex Hadamard matrices in order 6 beyond existing incomplete results.
• To enable the construction of larger families in higher dimensions, such as an 11-parameter family in 12 dimensions.
• To analyze degeneracy limits in Möbius transformations without breaking equivalence under row/column permutations.

Proposed method

• The paper uses the standard dephased form of $H_2$-reducible matrices, where all $2\times2$ submatrices are Hadamard.
• It applies unitarity constraints to derive relations between matrix blocks, leading to expressions involving matrices $A$ and $B$ defined via a self-adjoint, unitary $2\times2$ matrix $\Lambda$.
• The parameters are parameterized through Möbius transformations $\mathcal{M}_A$ and $\mathcal{M}_B$, with $z_1, z_2, z_3, z_4$ as complex variables on the unit circle.
• The construction uses the identity $A = F_2(-\frac{1}{2}e + i\frac{\sqrt{3}}{2}\Lambda)$, $B = F_2(-\frac{1}{2}e - i\frac{\sqrt{3}}{2}\Lambda)$, ensuring unitarity and unimodularity.
• Degeneracy in Möbius maps is handled by analyzing limit cases, showing equivalence of resulting matrices under row and column permutations.
• The method allows construction of a full three-parameter family even when transformations degenerate, preserving equivalence.

Experimental results

Research questions

• RQ1 Can a complete three-parameter family of $H_2$-reducible complex Hadamard matrices of order 6 be constructed that includes all known one- and two-parameter families as subfamilies?
• RQ2 How do degeneracies in Möbius transformations affect the resulting Hadamard matrices, and are the limits equivalent under matrix equivalence?
• RQ3 What is the behavior of the three-parameter family as it approaches the doubly degenerate point at $\theta = 0$?
• RQ4 Does the three-parameter family lead to a larger family in higher dimensions, such as 12 dimensions?
• RQ5 Can the limit behavior when $\theta \to 0$ recover known families like $F_6^{(2)}$ or $(F_6^{(2)})^T$?

Key findings

• The paper constructs a complete three-parameter family of $H_2$-reducible complex Hadamard matrices of order 6, encompassing all previously known one- and two-parameter families as subfamilies.
• The family is parameterized via Möbius transformations of the complex parameters $z_1, z_2, z_3, z_4$ on the unit circle, with the matrix structure determined by unitary and self-adjoint $2\times2$ matrices.
• Degeneracy in Möbius maps does not break the construction, as different limit paths yield equivalent matrices under row and column permutations.
• In the limit $\theta \to 0$, the family recovers the Fourier family $F_6^{(2)}$ when $z_1$ is used as the independent parameter, and a subfamily of $(F_6^{(2)})^T$ when $z_3$ is used.
• The three-parameter family enables the construction of an 11-parameter family of complex Hadamard matrices in 12 dimensions, the largest such family known so far.
• The family includes all known $H_2$-reducible matrices of order 6, and the absence of $H_2$-reducibility in $S_6^{(0)}$ confirms its non-inclusion.