On codes over quaternion integers

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 Ozen and Guzeltepe (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.