Decoding of cyclic codes over quaternion integers by modified Berlekamp–Massey algorithm

In this article, authors describe the decoding of cyclic codes over Hamilton quaternion integers (QI) of quaternion Mannheim weight one up to four coordinates for the correction of errors by modified Berlekamp–Massey algorithm (MBMA) of length \(n=\varphi \left(p\right)=p-1\) $n=\phi \left(p\right)=p-1$ and length \(2n- 1=2\varphi \left(p\right)- 1=2p-3\) $2n-1=2\phi \left(p\right)-1=2p-3$ , where \(p\) $p$ is a prime integer and \(\varphi \) $\phi$ is Euler phi function. Furthermore, these codes tradeoff error correction capability and code rates. Thus, the length of the codes over quaternion integers increased along with their better code rate and error correction capability.