Self-dual codes over a family of local rings
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Self‑dual codes over a family of local rings Steven T. Dougherty1 · Cristina Fernández‑Córdoba2 · Roger Ten‑Valls2 Received: 22 July 2019 / Accepted: 21 September 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We construct an infinite family of commutative rings Rq,𝛥 and we study codes over these rings as well as the structure of the rings. We define a canonical Gray map from Rq,𝛥 to vectors over the residue finite field of q elements and use it to relate codes over Rq,𝛥 to codes over the finite field 𝔽q . Finally, we determine the parameters for when self-dual codes exist and give various constructions for self-dual codes over Rq,𝛥. Keywords Self-dual codes · Codes over rings · Local rings Mathematics Subject Classification 94B05 · 11T71
1 Introduction In [8], a family of rings with characteristic 2 was introduced and it was shown how to construct quasi-cyclic codes over finite fields from cyclic codes over this family of rings. The importance of this result is that it gave an algebraic structure to quasi-cyclic codes to aid in their construction and classification. This work was expanded in [9], where it was shown that using G-codes over this family of rings, namely codes corresponding to ideals in group rings, that quasi-G-codes could be This work has been partially supported by the Spanish MINECO Grants TIN2013-40524-P and MTM2015-69138-REDT, and by the Catalan AGAUR Grant 2014SGR-691. * Steven T. Dougherty [email protected] Cristina Fernández‑Córdoba [email protected] Roger Ten‑Valls [email protected] 1
Department of Mathematics, University of Scranton, Scranton, PA 18510, USA
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Department of Information and Communications Engineering, Universitat Autònoma de Barcelona, 08193 Bellaterra, Spain
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constructed. As such, codes over this family of rings are an important family of codes. We shall investigate this family of rings, expanding its definition significantly to characteristic any prime, and study codes over these rings with an eye to constructing quasi-cyclic and quasi G-codes over the base field. This family of rings is a generalization of such rings such as Ak which were studied in [1, 2] and rings that were studied in [3–5]. More directly it is a generalization of the family of rings Rk where were studied extensively in such paper as [10–13]. In this paper, we shall make a unified approach to study these families of rings by making a broad generalization of the family of rings that we first studied in [8]. In that paper, this family of rings was first defined using only 𝔽2 rather than 𝔽q , as is done in this paper, as the base finite field. The rings in this paper are, therefore, a large family of rings which contains the various families of rings described in the references as subfamilies. We determine a linear Gray map from this family of rings to codes over finite fields and use it to relate these codes over these rings to codes over a finite field. In this paper, we give a natural way to use
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