Construction Of Linear Codes Over The Galois Ring GR(2³) With Hamming Distance

Abstract

This study focuses on the construction and analysis of linear codes over a Galois ring with eight elements, motivated by the need to develop error correcting codes beyond finite fields. The objective is to examine how the selection of generator vectors influences the minimum Hamming distance and the resulting error detection and correction capabilities. The methodology involves constructing two linear codes of length four and dimension two using different generator matrices. Codewords are generated through linear combinations of generator vectors, and the minimum Hamming distance is determined by evaluating the weights of all nonzero codewords. The results show that the first generator matrix produces a minimum distance of three, allowing the detection of up to two errors and correction of one error, while the second produces a minimum distance of two, allowing only single-error detection. The findings indicate that code performance is primarily influenced by the linear relationships among generator vectors rather than solely by the presence of zero divisors. In conclusion, careful selection of generator vectors is essential for optimizing linear codes over Galois rings and improving their performance in digital communication systems.