Abstract
A code is said to be propelinear if its automorphism group contains a subgroup acting on its codewords regularly. A subgroup of the group \(GA(r,q)\) of affine transformations is said to be regular if it acts regularly on vectors of \(\mathbb{F}_q^r\). Every automorphism of a regular subgroup of the general affine group \(GA(r,q)\) induces a permutation on the cosets of the Hamming code of length \(\frac{q^r-1}{q-1}\). Based on this permutation, we propose a construction of \(q\)-ary propelinear perfect codes of length \(\frac{q^{r+1}-1}{q-1}\). In particular, for any prime \(q\) we obtain an infinite series of almost full rank \(q\)-ary propelinear perfect codes.
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Acknowledgment
The author is grateful to F.I. Solov’eva for discussions in which a part of claims and approaches of this paper were found, and also to a reviewer for valuable remarks and suggestions leading to an improvement in the presentation.
Funding
The research was carried out at the expense of the Russian Science Foundation, project no. 22-21-00135, https://rscf.ru/en/project/22-21-00135/
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Translated from Problemy Peredachi Informatsii, 2022, Vol. 58, No. 1, pp. 65–79 https://doi.org/10.31857/S0555292322010041.
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Mogilnykh, I. On q-ary Propelinear Perfect Codes Based on Regular Subgroups of the General Affine Group. Probl Inf Transm 58, 58–71 (2022). https://doi.org/10.1134/S0032946022010045
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DOI: https://doi.org/10.1134/S0032946022010045