SKEW-CYCLIC LINEAR CODES OVER THE FINITE RING R_p = F_p [v_1,v_2,···,v_τ ]/∶ AN IN-DEPTH EXPLORATION

SKEW-CYCLIC LINEAR CODES OVER THE FINITE RING R_p = F_p [v_1,v_2,···,v_τ ]/<v_i^2 = 1,v_i v_j-v_j v_i>∶ AN IN-DEPTH EXPLORATION

K. Chatouh

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Abstract

This article introduces novel advancements in the realm of linear codes over the ring of integers modulo a prime, denoted as Rp = Fp[v1, v2, · · · , vτ ]/ v2 i = 1, vivj − vjvi  , with τ ≥ 1, p = qs and q is an odd prime. Specifically, we present a new Gray map and Gray images tailored for linear codes over Rp, facilitating efficient representation and manipulation of these codes. Building upon this foundation, the study delves into the characterization and properties of skew cyclic codes over Rp, a class of linear codes with intriguing mathematical structures. The investigation of skew cyclic linear code properties reveals new insights into their algebraic properties. This work not only contributes to the theoretical understanding of linear and skew cyclic codes over Rp but also suggests practical implications for coding theory.

Keywords

Linear codes, Skew cyclic codes, Gray map, Lee weight, Skew cyclic LCD codes.