MacDonald codes over the ring $${\mathbb {F}}_{p}+v{\mathbb {F}}_{p}+v^2{\mathbb {F}}_{p}$$
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(2019) 38:169
MacDonald codes over the ring Fp + vFp + v 2 Fp Yongkang Wang1 · Jian Gao1,2 Received: 15 October 2018 / Revised: 11 February 2019 / Accepted: 24 September 2019 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2019
Abstract In this paper, we consider MacDonald codes over the finite non-chain ring F p + vF p + v 2 F p and their applications in constructing secret sharing schemes and association schemes, where p is an odd prime and v 3 = v. We give some structural properties of MacDonald codes first. Then, we study the weight enumerators of torsion codes of these MacDonald codes. As some applications, constructing secret sharing schemes and association schemes is also investigated. Keywords MacDonald codes · Torsion codes · Secret sharing schemes · Association schemes Mathematics Subject Classification 94B05 · 11T71
1 Introduction MacDonald codes are a class of linear codes with two nonzero weights. Two weights linear codes have many wide applications in authentication codes, association schemes and secret sharing schemes. Two weights codes are also closely related to objects in different areas of mathematics such as strongly regular graphs, partial geometries, and projective point sets. Therefore, the construction of two weights linear codes has become a hot topic of coding theory, such as Shi et al. constructed some two weights projective Z4 -codes in Shi et al. (2017b) gave two new families of two weights codes by codes over finite non-chain rings in Shi et al. (2017a).
Communicated by Thomas Aaron Gulliver.
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Jian Gao [email protected]; [email protected] Yongkang Wang [email protected]
1
School of Mathematics and Statistics, Shandong University of Technology, Zibo 255000, People’s Republic of China
2
School of Mathematics and Statistics, Changsha University of Science and Technology, Changsha 410114, People’s Republic of China
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MacDonald codes have such good properties, so they have attracted the attention of coding scholars. The binary MacDonald codes were introduced in MacDonald (1960). And MacDonald codes over finite field Fq were studied in Patel (1975). In 2003, Colbourn and Gupta obtained two families of MacDonald codes over the ring Z4 from Z4 -simplex codes of types α and β (see Colbourn and Gupta 2003). Dertli and Cengellenmis (2011) studied the MacDonald codes over the finite non-chain ring F2 + vF2 with v 2 = v. In 2016, Wang et al. studied MacDonald codes over F p + vF p with v 2 = v. They also determined the access structure of secret sharing schemes based on these codes (see Wang et al. 2016). In Delsarte (1973), the association schemes approach was firstly used to deal with a collection of topics involving the weight distribution of a code. Association schemes are closely related to coding theory, graph theory and finite fields theory. Especially, they provide a framework to study codes and designs. Luo et al. (2018) constructed a class of linear codes with two weights over Fq by linear codes over t
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