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A New Non-Affine Family in the Karlsson-Szollosi Intersection
Abstract
We show that K6(2) ∩ X6(2) = D6(1) ∪ C6(1) up to standard Hadamard equivalence, where C6(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 D6(1). Our new parametrisation of the Szöllősi family X6(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.
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BibTeX
@misc{ranchin2026nonaffine,
title = {{A New Non-Affine Family in the Karlsson-Szollosi Intersection}},
author = {André Ranchin},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.33207},
eprint = {2610.33207},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.33207},
}
RIS
TY - UNPB
TI - A New Non-Affine Family in the Karlsson-Szollosi Intersection
AU - André Ranchin
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2610.33207
UR - https://arxiv.org/abs/2610.33207
AB - We show that K_6^(2) ∩ X_6^(2) = D_6^(1) ∪ 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.
ER -