Abstract
Decoding algorithms for the correction of errors for cyclic codes over quaternion integers of quaternion Mannheim weight one up to two coordinates are discussed by Özen and Güzeltepe (Eur J Pure Appl Math 3(4):670–677, 2010). Though, Neto et al. (IEEE Trans Inf Theory 47(4):1514–1527, 2001) proposed decoding algorithms for the correction of errors of arbitrary Mannheim weight. In this study, we followed the procedures used by Neto et al. and suggest a decoding algorithm for an \(n\) length cyclic code over quaternion integers to correct errors of quaternion Mannheim weight two up to two coordinates. Furthermore, we establish that; over quaternion integers, for a given \(n\) length cyclic code there exist a cyclic code of length \(2n-1\). The decoding algorithms for the cyclic code of length \(2n-1\) are given, which correct errors of quaternion Mannheim weight one and two. In addition, we show that the cyclic code of length \(2n-1\) is maximum-distance separable (MDS) with respect to Hamming distance.
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Shah, T., Rasool, S.S. On codes over quaternion integers. AAECC 24, 477–496 (2013). https://doi.org/10.1007/s00200-013-0203-2
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DOI: https://doi.org/10.1007/s00200-013-0203-2