A New Non-Affine Family in the Karlsson-Szollosi Intersection

Abstract

We show that $K_6^{(2)} \cap X_6^{(2)} = D_6^{(1)} \cup C_6^{(1)}$ up to standard Hadamard equivalence, where $C_6^{(1)}$ is a new non-affine one-parameter family of complex Hadamard matrices. This demonstrates that the intersection of the two-parameter Hadamard families of Karlsson and Szöllősi is strictly larger than the known affine Diţă curve $D_6^{(1)}$. Our new parametrisation of the Szöllősi family $X_6^{(2)}$ in Karlsson coordinates and our constructive classification of the Karlsson-Szöllősi intersection shed light on the intersection of block-reducible forms and hypocycloid constraints. This may support the search for mutually unbiased bases in six dimensions.

The paper determines the overlap, up to standard Hadamard equivalence, between Karlsson’s and Szöllősi’s two-parameter families of order-six complex Hadamard matrices, $K_6^{(2)}$ and $X_6^{(2)}$. The known affine Diţă curve $D_6^{(1)}$ lies in both; the question is whether it exhausts their intersection. The main result is $K_6^{(2)}\cap X_6^{(2)}=D_6^{(1)}\cup C_6^{(1)}$, where $C_6^{(1)}$ is a new non-affine one-real-parameter family. An explicit dephased representative is constructed from unit phases $u,v$ satisfying $|(1+u)(1+v)|=2$. In angular coordinates this becomes $\cos(\alpha/2)\cos(\beta/2)=1/2$. The paper proves non-affineness and membership in both parent families. The new family is distinct from the Diţă curve but intersects it at the Diţă matrix. The construction uses an explicit parametrisation of Szöllősi matrices in Karlsson block coordinates, with three permutation charts handling coordinate degeneracies. Completeness is established by enumerating all $225$ independent row-pair and column-pair decompositions into $2\times2$ blocks. These comprise three generic frames, $168$ algebraically empty conditional frames, $27$ Diţă frames, and $27$ further conditional frames assigned to $C_6^{(1)}$. The conditional classification is computer-assisted and exact, using polynomial ideals, Gröbner-basis reductions, and explicit equivalence certificates. The scope is the intersection of these particular order-six families, not the full Hadamard space. Connections to mutually unbiased bases in dimension six and to higher composite dimensions remain motivations and possible applications; no new mutually unbiased configuration or existence bound is established.