Abstract
In this paper we investigate left ideals as codes in twisted skew group rings. The considered rings, which are in most cases algebras over a finite field, allow us to retrieve many of the well-known codes. The presentation, given here, unifies the concept of group codes, twisted group codes and skew group codes.
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Notes
Here, \(\textrm{Aut}(K)\) denotes the automorphism group of the field K.
Indeed, there exists a diagonal matrix D such that \({\widetilde{g}}=(\Theta _{g},D\cdot g)\). Then we have \({\widetilde{g}}(\lambda e_h)= D^{\Theta _{g^{-1}}} \cdot g^{\Theta _{g^{-1}}}(\lambda e_h)^{\Theta _{g^{-1}}}=\lambda ^{\Theta _{g^{-1}}}D^{\Theta _{g^{-1}}} \cdot g^{\Theta _{g^{-1}}}(e_h)^{\Theta _{g^{-1}}}=\Theta _g(\lambda )({\widetilde{g}}e_h)\).
References
Batoul A., Boucher D., Boulanouar R.: On overview of skew constacyclic codes and their subclasses of LCD codes. Adv. Math. Commun. 15(4), 611–632 (2021).
Bazzi L.M., Mitter S.K.: Some randomized code constructions from group actions. IEEE Trans. Inf. Theory 52(7), 3210–3219 (2006).
Behajaina A.: On BCH split metacyclic codes. Adv. Math. Commun. 17, 1181 (2021).
Berman S.D.: Semisimple cyclic and Abelian codes. II. Kibernetika (Kiev) 3, 21–30 (1967) (Russian).
Bernal J.J., del Río A., Simón J.J.: An intrinsical description of group codes. Des. Codes Cryptogr. 51, 289–300 (2009).
Borello M., de la Cruz J., Willems W.: On checkable group codes. J. Algebra Appl. 21, 2250125 (2022).
Borello M., Jamous A.: Dihedral codes with prescribed minimum distance. In: Bajard J.C., Topuzoglu A. (eds.) Arithmetic of Finite Fields. WAIFI 2020. Lecture Notes in Computer Science, vol. 12542. Springer, Cham (2021).
Borello M., Willems W.: Group codes over fields are asymptotically good. Finite Fields Appl. 68, 101738 (2020).
Borello M., Willems W., Zini G.: On ideals in group algebras: an uncertainty principle and the Schur product. Forum Math. (2022). https://doi.org/10.1515/forum-2022-0064.
Boucher D., Geiselmann W., Ulmer F.: Skew-cyclic codes. Appl. Algebra Eng. Commun. Comput. 18, 379–389 (2007).
de la Cruz J., Willems W.: On group codes with complementary duals. Des. Codes Cryptogr. 86, 2065–2073 (2018).
de la Cruz J., Willems W.: Twisted group codes. IEEE Trans. Inf. Theory 67, 5178–5184 (2021).
Dougherty S.T., Sahinkaya S., Yıldız B.: Skew G-codes. J. Algebra Appl. 22, 2350056 (2021).
Ellis G., Kholodna I.: Computing second cohomology of finite groups with trivial coefficients. Homol. Homotopy Appl. 1(1), 163–168 (1999).
Grassl M.: Bounds on the minimum distance of linear codes. Online http://www.codetables.de.
Huffman C.W., Pless V.: Funadamentals of Error Correcting Codes. Cambridge University Press, Cambridge (2003).
Karpilovsky G.: Projective Representations of Finite Groups. Marcel Dekker, New York (1985).
Linckelmann M.: The Block Theory of Finite Group Algebras. London Math. Soc., Student Text, vol. 91, Cambridge University Press, Cambridge (2018).
MacWilliams F.J.: Codes and ideals in group algebras. In: Bose R.C., Dowling T.A. (eds.) Combinatorial Mathematics and its Applications, Proceedings, pp. 317–328 (1967).
Morandi P.: Field and Galois Theory. Graduate Texts in Mathematics, vol. 167, p. xvi. Springer, New York (1996).
Passman D.S.: The Jacobson radical of group rings of locally finite groups. Trans. Am. Math. Soc. 349, 4693–4751 (1997).
Prange E.: Cyclic Error-Correcting Codes in Two Symbols. Air Force Cambridge Research Center, Cambridge, MA, Tech. Rep. AFCRC-TN-57-103 (1957).
Ran M., Snyders J.: A cyclic \([6,3,4]\) group code and the hexacode over \({\rm GF}(4)\). IEEE Trans. Inf. Theory 42, 1250–1253 (1996).
Willems W.: Codes in group algebras. In: Huffman W.C., Kim J.-L., Solé P. (eds.) Concise Encyclopedia of Coding Theory, Chap, pp. 363–384. 16. CRC Press, Chapman and Hall, Boca Raton (2021).
Willems W.: Codes in twisted group algebras (To appear).
Wood Y.A.: Duality for modules over finite rings and applications to coding theory. Am. J. Math. 121, 555–575 (1999).
Acknowledgements
The first author is grateful for the support of the Israel Science Foundation (grant no. 353/21). The second author was partially supported by the ANR-21-CE39-0009 - BARRACUDA (French Agence Nationale de la Recherche). The third author was partially supported by Fundación Banco de la República under project 4649. We also thank the anonymous referees for valuable comments and suggestions.
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Communicated by K.-U. Schmidt.
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Behajaina, A., Borello, M., Cruz, J.d.l. et al. Twisted skew G-codes. Des. Codes Cryptogr. (2024). https://doi.org/10.1007/s10623-024-01367-0
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DOI: https://doi.org/10.1007/s10623-024-01367-0