Application of MacWilliams' Theorem for Complete Weight Enumerators on Galois Fields
Abstract
This paper implements the extended MacWilliams theorem for linear codes, revisiting the MacWilliams theorem for the Complete Weight Enumerator (CWE) of codes with q = 3, 4, 5, and 7. Linear codes are a mathematical concept that can be described through the distribution of weights in each codeword. CWE serves to provide a complete representation of the symbols in each codeword. This article discusses how the MacWilliams theorem can be efficiently used to calculate the CWE of codes and optimize the design of error-correcting codes. The results of this research include the calculation of CWE for and . This study contributes theoretically to the understanding of linear code structures while opening up opportunities for the development of more efficient error-correcting code algorithms in modern communication systems.
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