A Novel Approach to Constructing Certain Classes of Quasi-Cyclic Codes

This paper investigates a construction algorithm for two specific families of quasi-cyclic codes defined over a finite commutative chain ring. First, by employing the Generalized Discrete Fourier Transform, we develop an efficient and systematic algorithm for constructing the generator matrix of repeated-root quasi-cyclic codes under specific structural constraints on the code length and the underlying ring. The proposed method avoids the need for exhaustive enumeration of constituent subcodes and instead operates directly on their generator matrices, leading to improved computational performance. Building on this result, using the Discrete Fourier Transform, we further specialize the proposed framework to derive an algorithm for obtaining the generator matrix of a particular class of simple-root quasi-cyclic codes over a restricted and well-defined category of rings. This specialization demonstrates the performance of the proposed approach and highlights its applicability to different quasi-cyclic code structures in a unified algebraic setting. The proposed construction methods offer significant improvements in computational efficiency when compared to existing approaches that rely on code construction techniques based on constituent subcodes. To evaluate the practical benefits of the proposed algorithms, we present a detailed performance comparison in terms of encoding time per iteration and memory consumption. The comparative results, illustrated through histograms, clearly indicate that the proposed methods achieve faster encoding, lower memory usage, highlighting their superior performance compared to conventional construction techniques as the code length increases.