Linear Code Analysis over GR(9) Using Hamming Distance - Ferry Prabowo; Santoso Budi Wiyono; Putranto Hadi Utomo (International Journal of Interdisciplinary Research, 2026)
Data transmission in digital communication systems is vulnerable to disturbances such as noise and interference, which may cause errors in the received information. Therefore, coding mechanisms are required to detect and correct such errors. This study investigates the construction of linear codes over the Galois ring of nine elements . The code is constructed as a submodule of length 4 with dimension 2 , meaning that all codewords are formed as linear combinations of two linearly independent generator vectors. Two generator matrices are employed to analyze the effect of generator structure on code performance. All generated codewords are computed and evaluated using Hamming weight and Hamming distance to determine the minimum distance. The results show that the code generated by the first generator matrix has a minimum distance 3, allowing it to detect up to two errors and correct one error. In contrast, the second generator matrix produces a code with minimum distance 2 , which can only detect a single error without a correction capability. This difference indicates that code performance is more influenced by the linear relationships among generator vectors than by the presence of zero divisors in the ring structure. This study highlights the importance of selecting appropriate generator matrices in constructing linear codes over finite rings and demonstrates the potential of Galois rings as an alternative framework in coding theory.