On Z_p^rZ_p^rZ_p^s-Additive Cyclic Codes

2022/02/23 by Fernández-Córdoba, Cristina, Pathak, Sachin, Upadhyay, Ashish Kumar
#Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #Information Theory (cs.IT)

paper · doi:10.48550/arxiv.2202.11454

Abstract

In this paper, we introduce ℤ_p^rℤ_p^rℤ_p^s-additive cyclic codes for r≤ s. These codes can be identified as ℤ_p^s[x]-submodules of ℤ_p^r[x]/⟨ x^α-1⟩ × ℤ_p^r[x]/⟨ x^β-1⟩× ℤ_p^s[x]/⟨ x^γ-1⟩. We determine the generator polynomials and minimal generating sets for this family of codes. Some previous works has been done for the case p=2 with r=s=1, r=s=2, and r=1,s=2. However, we show that in these previous works the classification of these codes were incomplete and the statements in this paper complete such classification. We also discuss the structure of separable ℤ_p^rℤ_p^rℤ_p^s-additive cyclic codes and determine their generator polynomials. Further, we also study the duality of ℤ_p^s[x]-submodules. As applications, we present some examples and construct some optimal binary codes.

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