Geodesic
Relationship between Codes and Idempotents in a Dihedral Group Algebra
Matematičeskie zametki, Tome 107 (2020) no. 2, pp. 178-194 Cet article a éte moissonné depuis la source Math-Net.Ru

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Codes in the dihedral group algebra $\mathbb{F}_qD_{2n}$, i.e., left ideals in this algebra, are studied. A generating idempotent is constructed for every code in $\mathbb{F}_qD_{2n}$ given by its image under the Wedderburn decomposition of this algebra. By using a selected set of idempotents, the inverse Wedderburn transform for the algebra $\mathbb{F}_qD_{2n}$ is constructed. The image of some codes under the Wedderburn decomposition is described directly in terms of their generating idempotents. Examples of the application of the obtained results to induced codes are considered.
Keywords: dihedral group, idempotents, Wedderburn decomposition, noncommutative codes.
Mots-clés : group algebras 
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K. V. Vedenev; V. M. Deundyak. Relationship between Codes and Idempotents in a Dihedral Group Algebra. Matematičeskie zametki, Tome 107 (2020) no. 2, pp. 178-194. http://geodesic.mathdoc.fr/item/MZM_2020_107_2_a1/

[1] R. J. McEliece, Public-Key Cryptosystem Based on Algebraic Coding Theory, DSN Progress Report No 42-44, 1978

[2] D. J. Bernstein, “Grover vs. McEliece”, Post-Quantum Cryptography, Lecture Notes in Comput. Sci., 6061, Springer, Berlin, 2010, 73–80 | DOI | MR | Zbl

[3] V. M. Deundyak, Yu. V. Kosolapov, “On the Berger–Loidreau cryptosystem on the tensor product of codes”, J. Comput. Eng. Math., 5:2 (2018), 16–33 | MR | Zbl

[4] K. Garsia-Pilyado, S. Gonsales, V. T. Markov, K. Martines, “Neabelevy gruppovye kody nad proizvolnym konechnym polem”, Fundament. i prikl. matem., 20:1 (2015), 17–22 | MR

[5] E. Kouselo, S. Gonsales, V. T. Markov, K. Martines, A. A. Nechaev, “Predstavleniya kodov Rida–Solomona i Rida–Mallera idealami”, Algebra i logika, 51:3 (2012), 297–320 | MR | Zbl

[6] C. Polcino Milies, F. D. de Melo, “On cyclic and Abelian codes”, IEEE Trans. Inform. Theory, 59:11 (2013), 7314–7319 | DOI | MR | Zbl

[7] V. M. Deundyak, Yu. V. Kosolapov, “Algoritmy dlya mazhoritarnogo dekodirovaniya gruppovykh kodov”, Model. i analiz inform. sistem, 22:4 (2015), 464–482 | DOI | MR

[8] K. V. Vedenev, V. M. Deundyak, “Kody v diedralnoi gruppovoi algebre”, Model. i analiz inform. sistem, 25:2 (2018), 232–245 | DOI | MR

[9] F. E. Brochero Martínez, “Structure of finite dihedral group algebra”, Finite Fields Appl., 35 (2015), 204–214 | DOI | MR | Zbl

[10] V. M. Deundyak, Yu. V. Kosolapov, “Kriptosistema na indutsirovannykh gruppovykh kodakh”, Model. i analiz inform. sistem, 23:2 (2016), 137–152 | DOI | MR

[11] C. Polcino Milies, S. K. Sehgal, An Inroduction to Group Rings, Kluwer Acad. Publ., Dordrecht, 2002 | MR