Abstract
After recalling the definition of some codes as modules over skew polynomial rings, whose multiplication is defined by using an endomorphism and a derivation, and some basic facts about them, in the first part of this paper we study some of their main algebraic and geometric properties. Finally, for module skew codes constructed only with an automorphism, we give some BCH type lower bounds for their minimum distance.
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References
Bosma W., Cannon J., Playoust C.: The Magma algebra system. I. The user language. J. Symb. Comput. 24(3–4), 235–265 (1997).
Boucher D., Ulmer F.: Linear codes using skew polynomials with automorphisms and derivations. Des. Codes Cryptogr. 70(3), 405–431 (2014).
Boulagouaz M.H., Leroy A.: \((\sigma,\delta )\)-codes. Adv. Math. Commun. 7(4), 463–474 (2013).
Cohn P.M.: Free Rings and Their relations. London Mathematical Society. Academic Press, London (1971).
Hartmann C.R., Tzeng K.K.: Generalizations of the BCH bound. Inf. Control 20, 489–498 (1972).
Herstein I.N.: Topics in Algebra, 2nd edn. Xerox College Publishing, Lexington (1975).
Jacobson N.: Pseudo-linear transformations. Ann. Math. 38(2), 484–507 (1937).
Jacobson N.: The Theory of Rings. American Mathematical Society Mathematical Surveys, vol. II. American Mathematical Society, New York (1943).
Lam T.Y., Leroy A.: Vandermonde and Wronskian matrices over division rings. J. Algebra 119, 308–336 (1988).
Leroy A.: Pseudo-linear transformation and evaluation in Ore extension. Bull. Belg. Math. Soc. 2, 321–345 (1995).
Leroy A.: Noncommutative polynomial maps. J. Algebra Appl. 11, 1250076 (2012).
Radkova D., Van Zanten A.J.: Constacyclic codes as invariant subspaces. Linear Algebra Appl. 430(2–3), 855–864 (2009).
Tapia Cuitiño L.F., Tironi A.L.: Dual codes of product semi-linear codes. Linear Algebra Appl. 457, 114–153 (2014).
Acknowledgements
The authors would like to thank the referees for their careful reading and for many useful suggestions which improved the final presentation of the paper. This work is in the framework of the Project Anillo ACT 1415 PIA CONICYT and the second author was partially supported by Proyecto VRID N. 214.013.039-1.OIN.
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Communicated by C. Mitchell.
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Tapia Cuitiño, L.F., Tironi, A.L. Some properties of skew codes over finite fields. Des. Codes Cryptogr. 85, 359–380 (2017). https://doi.org/10.1007/s10623-016-0311-7
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DOI: https://doi.org/10.1007/s10623-016-0311-7