On the error-detecting capability of the linear quasigroup code

Abstract

In this paper we consider an error-detecting code based on linear quasigroups. Namely, each input block $a_0a_1\ldots a_{n-1}$ is extended into a block $a_0a_1\ldots a_{n-1}d_0d_1\ldots d_{n-1}$, where the redundant characters $d_0, d_1, \ldots , d_{n-1}$ are defined with $d_i=a_i*a_{i+1}*a_{i+2}$, where $*$ is a linear quasigroup operation and the operations in the indexes are modulo n. We give a proof that under some conditions the code is linear. Using this fact, we contribute to the determination of the error-detecting capability of the code. Namely, we determine the Hamming distance of the code and from there we obtain the number of errors that the code will detect for sure when linear quasigroups of order 4 from the best class of quasigroups of order 4 for which the constant term in the linear representation is zero matrix are used for coding. All results in the paper are derived for arbitrary length of the input blocks. With the obtained results we showed that when a small linear quasigroup of order 4 from the best class of quasigroups of order 4 is used for coding, the number of errors that the code surely detects is upper bounded with 4.