Construction of all MDS and involutory MDS matrices
<|reference_start|>Construction of all MDS and involutory MDS matrices: In this paper, we propose two algorithms for a hybrid construction of all $n\times n$ MDS and involutory MDS matrices over a finite field $\mathbb{F}_{p^m}$, respectively. The proposed algorithms effectively narrow down the search space to identify $(n-1) \times (n-1)$ MDS matrices, facilitating the generation of all $n \times n$ MDS and involutory MDS matrices over $\mathbb{F}_{p^m}$. To the best of our knowledge, existing literature lacks methods for generating all $n\times n$ MDS and involutory MDS matrices over $\mathbb{F}_{p^m}$. In our approach, we introduce a representative matrix form for generating all $n\times n$ MDS and involutory MDS matrices over $\mathbb{F}_{p^m}$. The determination of these representative MDS matrices involves searching through all $(n-1)\times (n-1)$ MDS matrices over $\mathbb{F}_{p^m}$. Our contributions extend to proving that the count of all $3\times 3$ MDS matrices over $\mathbb{F}_{2^m}$ is precisely $(2^m-1)^5(2^m-2)(2^m-3)(2^{2m}-9\cdot 2^m+21)$. Furthermore, we explicitly provide the count of all $4\times 4$ MDS and involutory MDS matrices over $\mathbb{F}_{2^m}$ for $m=2, 3, 4$.<|reference_end|>
@article{kumar2024construction,
    title={Construction of all MDS and involutory MDS matrices},
    author={Yogesh Kumar, P.R.Mishra, Susanta Samanta, Kishan Chand Gupta and Atul
  Gaur},
    journal={Advances in Mathematics of Communications 2024},
    year={2024},
    doi={10.3934/amc.2024033},
    archivePrefix={arXiv},
    eprint={2403.10372},
    primaryClass={cs.CR}
}