Minimum Distance Computation Efficiency in Module Codes over Finite Chain Rings - Meishrin, Ishlahrahmi; Sylviani, Sisilia; Achiaa, Amma (Indonesian Journal of Applied Mathematics and Statistics, 2025)
In coding theory, the module structure can be used in the construction and analysis of codes for reliable data transmission. A code that is effective in detecting and correcting errors that occur during data transmission is a good code. The parameter to determine the reliability of the code in error detection and correction is by calculating its minimum distance. However, to calculate the minimum distance, it is necessary to evaluate all the Hamming weights of all non-zero code words, if the size of the code are very long, the calculation of the minimum distance will take a long time so it is not very efficient if we use this method. Therefore, the module structure of the ring, especially over the finite chain ring, can be used to simplify the calculation process of the minimum distance. The complexity of calculating the minimum distance can be reduced by viewing the code as a module and utilizing its modular structure. Previous research has shown that certain structural characteristics of module codes can significantly simplify this computation. In this article, we present an efficient method for determining the minimum distance of a module code defined over a finite chain ring. We show that it only required the hamming weight of the code generator to find the code’s minimum distance. This approach provide an efficient way to analyze the code’s reability and give algebraic insight about the code’s behaviour. Our article highlight the practical advantages of applying a module theory perspective to coding theory, which offers an efficient and theoretically grounded framework for evaluating and constructing error-correcting codes with strong performance guarantees.