We prove that cocyclic Hadamard matrices of order 8p, with <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$p > 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>></mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation> prime, can be described using an <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$8 \times 8$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>8</mn> <mo>×</mo> <mn>8</mn> </mrow> </math></EquationSource> </InlineEquation> block template array. In this framework, underlying cocycles are translated into signs and actions related to the blocks and their columns, respectively. Our method uses the result that if a cocyclic Hadamard matrix has an indexing group <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$G = K \ltimes N$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>G</mi> <mo>=</mo> <mi>K</mi> <mo>⋉</mo> <mi>N</mi> </mrow> </math></EquationSource> </InlineEquation>, where N is cyclic of odd order, then <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$H^2(K, A) \cong H^2(G, A)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>H</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi>K</mi> <mo>,</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mrow> <mo>≅</mo> <msup> <mi>H</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo>,</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> whenever A is a finite trivial G-module of order coprime to |N|. This result implies that cocyclic Hadamard matrices of order 8p exhibit no coboundary dependency, allowing for a more concise description of such matrices. Our findings generalise to certain complex cocyclic Hadamard matrices, and we also investigate cocyclic Hadamard matrices of orders 24 and 40.

A Block Structure for Cocyclic Hadamard Matrices of Order 8p

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发表日期：

2025年5月5日

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Santiago Barrera Acevedo,
Heiko Dietrich,
Kshitija Vaidya

摘要

We prove that cocyclic Hadamard matrices of order 8p, with $p > 3$ $p > 3$ prime, can be described using an $8 \times 8$ $8 \times 8$ block template array. In this framework, underlying cocycles are translated into signs and actions related to the blocks and their columns, respectively. Our method uses the result that if a cocyclic Hadamard matrix has an indexing group $G = K \ltimes N$ $G = K ⋉ N$ , where N is cyclic of odd order, then $H^2(K, A) \cong H^2(G, A)$ $H^{2} (K, A) ≅ H^{2} (G, A)$ whenever A is a finite trivial G-module of order coprime to |N|. This result implies that cocyclic Hadamard matrices of order 8p exhibit no coboundary dependency, allowing for a more concise description of such matrices. Our findings generalise to certain complex cocyclic Hadamard matrices, and we also investigate cocyclic Hadamard matrices of orders 24 and 40.