A generalisation of bent vectors for Butson Hadamard matrices
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915244767444992 |
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| author | Armario, José Andrés Egan, Ronan Kharaghani, Hadi Catháin, Padraig Ó |
| author_facet | Armario, José Andrés Egan, Ronan Kharaghani, Hadi Catháin, Padraig Ó |
| contents | An $n\times n$ complex matrix $M$ with entries in the $k^{\textrm{th}}$ roots of unity which satisfies $MM^{\ast} = nI_{n}$ is called a Butson Hadamard matrix. While a matrix with entries in the $k^{\textrm{th}}$ roots typically does not have an eigenvector with entries in the same set, such vectors and their generalisations turn out to have multiple applications. A bent vector for $M$ satisfies $M{\bf x} = λ{\bf y}$ where ${\bf x}$ has entries in the $k^{\textrm{th}}$ roots of unity and all entries of $\textbf{y}$ are complex numbers of norm $1$. Such a bent vector ${\bf x}$ is self-dual if ${\bf y} = μ{\bf x}$ and conjugate self-dual if ${\bf y} = μ\overline{\bf x}$ for some $μ$ of norm $1$.
Using techniques from algebraic number theory, we prove some order conditions and non-existence results for self-dual and conjugate self-dual bent vectors; using tensor constructions and Bush-type matrices we give explicit examples. We conclude with an application to the covering radius of certain non-linear codes generalising the Reed Muller codes. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_16579 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A generalisation of bent vectors for Butson Hadamard matrices Armario, José Andrés Egan, Ronan Kharaghani, Hadi Catháin, Padraig Ó Combinatorics Information Theory 05B20, 94B25, 94A60 An $n\times n$ complex matrix $M$ with entries in the $k^{\textrm{th}}$ roots of unity which satisfies $MM^{\ast} = nI_{n}$ is called a Butson Hadamard matrix. While a matrix with entries in the $k^{\textrm{th}}$ roots typically does not have an eigenvector with entries in the same set, such vectors and their generalisations turn out to have multiple applications. A bent vector for $M$ satisfies $M{\bf x} = λ{\bf y}$ where ${\bf x}$ has entries in the $k^{\textrm{th}}$ roots of unity and all entries of $\textbf{y}$ are complex numbers of norm $1$. Such a bent vector ${\bf x}$ is self-dual if ${\bf y} = μ{\bf x}$ and conjugate self-dual if ${\bf y} = μ\overline{\bf x}$ for some $μ$ of norm $1$. Using techniques from algebraic number theory, we prove some order conditions and non-existence results for self-dual and conjugate self-dual bent vectors; using tensor constructions and Bush-type matrices we give explicit examples. We conclude with an application to the covering radius of certain non-linear codes generalising the Reed Muller codes. |
| title | A generalisation of bent vectors for Butson Hadamard matrices |
| topic | Combinatorics Information Theory 05B20, 94B25, 94A60 |
| url | https://arxiv.org/abs/2412.16579 |