The quaternary complex Hadamard matrices of orders 10, 12, and 14
Pekka H. J. Lampio, Ferenc Sz\"oll\H{o}si, Patric R. J., \"Osterg{\aa}rd
TL;DR
This paper classifies quaternary complex Hadamard matrices of orders 10, 12, and 14, introduces a new parametrization scheme, and discusses the existence of parametric families and isolated matrices.
Contribution
It provides a complete classification for orders 10 and 12, introduces a new parametrization scheme, and demonstrates the existence of isolated matrices at order 14.
Findings
All 10x10 and 12x12 matrices belong to parametric families.
A new parametrization scheme for affine families is proposed.
Existence of an isolated 14x14 matrix shows limits of parametrization methods.
Abstract
A complete classification of quaternary complex Hadamard matrices of orders 10, 12 and 14 is given, and a new parametrization scheme for obtaining new examples of affine parametric families of complex Hadamard matrices is provided. On the one hand, it is proven that all 10x10 and 12x12 quaternary complex Hadamard matrices belong to some parametric family, but on the other hand, it is shown by exhibiting an isolated 14x14 matrix that there cannot be a general method for introducing parameters into these types of matrices.
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Taxonomy
Topicsgraph theory and CDMA systems · Mathematics and Applications · Coding theory and cryptography