New classes of codes over $$R_{q,p,m}={\mathbb {Z}}_{p^{m}}[u_{1}, u_{2}, \ldots , u_{ q}]/\left\langle u_{i}^{2}=0,u_{i
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New classes of codes over Rq,p,m = Zpm [u1 , u2 , . . . , uq ]/ ui2 = 0, ui uj = uj ui and their applications Karima Chatouh1 · Kenza Guenda2 · T. Aaron Gulliver3 Received: 30 August 2019 / Revised: 25 April 2020 / Accepted: 29 April 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020
Abstract In this paper, we consider the construction of new classes of linear codes over the ring Rq, p,m = Z pm [u 1 , u 2 , . . . , u q ]/ u i2 = 0, u i u j = u j u i for i = j and 1 ≤ i, j ≤ q. The simplex and MacDonald codes of types α and β are obtained over Rq, p,m . We characterize some linear codes over Z pm that are the torsion codes and Gray images of these simplex and MacDonald codes, and determine the minimal codes. Keywords Simplex codes · MacDonald codes · Secret sharing scheme · Linear codes · Gray images · Torsion codes Mathematics Subject Classification 11TXX · 11T71 · 14G50 · 15Axx · 15B33
1 Introduction Linear codes over finite rings are of significant interest because of their role in algebraic coding theory and applications in combined coding and modulation. Codes over rings have been examined extensively, e.g., Chatouh et al. (2017a, b) and Yildiz and Kelebek (2015). The simplex and MacDonald codes are two well-known classes of codes. The simplex and MacDonald codes over the ring Rq, p,m = Z pm [u 1 , u 2 , . . . , u q ]/u i2 = 0, u i u j = u j u i are introduced in this paper. The motivation for this is that the Gray images and torsion codes of
Communicated by Masaaki Harada.
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Karima Chatouh [email protected]
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Faculty of Mathematics and Informatics, Department of Mathematics, Mostefa Ben Boulaïd University, Batna 2, Batna, Algeria
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Faculty of Mathematics, USTHB, University of Science and Technology of Algiers, Algiers, Algeria
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Department of Electrical and Computer Engineering, University of Victoria, PO Box 1700, STN CSC, Victoria, BC V8W 2Y2, Canada 0123456789().: V,-vol
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these codes for m = 1 are minimal. Minimal codes are widely used in secret sharing schemes based on linear codes. Secret sharing schemes were independently introduced by Shamir and Blakely in 1979. These schemes are used in a number of applications and so is an important field of research. They include a dealer with a secret, a set of n − 1 parties, and a collection A of subsets of parties called the access structure. Many secret sharing schemes have been developed using linear error correcting codes (Beimel 2011; Chen et al. 2013; Ding et al. 1997; Ding and Salomaa 2006; Ding and Yuan 2003; Li et al. 2013; Massey 1993; Yuan and Ding 2006). In particular, secret sharing schemes have been obtained with an access structure characterized by the duality of linear codes (Karnin et al. 1983; Massey 1995). This paper considers linear codes over the ring Rq, p,m = Z pm [u 1 , u 2 , . . . , u q ]/u i2 = 0, u i u j = u j u i . The torsion codes and Gray images of the simplex and MacDonald codes of types α and β over this ring are constructed as
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