2022/02/02 by Patric R. J. Östergård · 1 citation
Computer Science · Engineering · Mathematics · #Coding theory and cryptography #Mathematics and Applications #graph theory and CDMA systems
paper · pdf · doi:10.1007/s10801-021-01109-8
openalex publication_date 2022/02/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
Abstract Two matrices H1 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>H</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math> and H2 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>H</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> with entries from a multiplicative group G are said to be monomially equivalent, denoted by H1≅ H2 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>H</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>≅</mml:mo> <mml:msub> <mml:mi>H</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> </mml:math> , if one of the matrices can be obtained from the other via a sequence of row and column permutations and, respectively, left- and right-multiplication of rows and columns with elements from G . One may further define matrices to be Hadamard equivalent if H1 ≅ φ (H2) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>H</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>≅</mml:mo> <mml:mi>ϕ</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>H</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> for some φ ∈ \mathrm Aut(G) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ϕ</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>Aut</mml:mi> <mml:mo>(</mml:mo> <mml:mi>G</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . For many classes of Hadamard and related matrices, it is straightforward to show that these are closed under Hadamard equivalence. It is here shown that also the set of Butson-type Hadamard matrices is closed under Hadamard equivalence.