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All Journal Media Statistika CAUCHY: Jurnal Matematika Murni dan Aplikasi JURNAL PENGABDIAN KEPADA MASYARAKAT BAREKENG: Jurnal Ilmu Matematika dan Terapan Jurnal Pengabdian kepada Masyarakat International Journal of Interdisciplinary Research
Santoso Budi Wiyono
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Journal : International Journal of Interdisciplinary Research

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Linear Code Analysis over GR(9) Using Hamming Distance Ferry Prabowo; Santoso Budi Wiyono; Utomo, Putranto Hadi
International Journal of Interdisciplinary Research Vol. 2 No. 2 (2026): Vol 2 no 2 July 2026
Publisher : Ponpes As-Salafiyyah Asy-Syafi'iyyah

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.71305/ijir.v2i2.1623

Abstract

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.
Construction Of Linear Codes Over The Galois Ring GR(2³) With Hamming Distance Arif Febrorianto Hidayat; Putranto Hadi Utomo; Santoso Budi Wiyono
International Journal of Interdisciplinary Research Vol. 2 No. 2 (2026): Vol 2 no 2 July 2026
Publisher : Ponpes As-Salafiyyah Asy-Syafi'iyyah

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.71305/ijir.v2i2.1626

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.
Co-Authors Arif Febrorianto Hidayat Ekasiwi, Chessa Fanny Ferry Prabowo Mika Asrini Putranto Hadi Utomo Putranto Hadi Utomo Rahayu, Eka Widia S Siswanto Siswanto Siswanto Siswanto Sufia Nur Janah Supriyadi Wibowo Sutrima Sutrima Sutrima, Sutrima Sutrima Vika Yugi Kurniawan Vika Yugi Kurniawan Wibowo, Supriyadi Wibowo Winita Sulandari Yulia Indah Puspitasari Zyuhri, Syaifudin