Theoretical proof for the number of errors that one linear code detects/corrects when linear quasigroups of order 4 are used for coding

Abstract

In this paper we will analyze one linear code from the theoretical point of view. Namely, the code definition is based on linear quasigroups. In the previous work we classified the quasigroups of order 4 according to their probability of undetected errors. Now, in this paper we will conclude whether the linear quasigroups that are in the same class in this classification obtain equal number of surely detected incorrectly transmitted bits. Also, we will classify the linear quasigroups of order 4 according to the number of errors that the code surely detects when they are used for coding. At the end we will make conclusion which quasigroups of order 4 are overall best for coding having in mind both important parameters for every code for error detection: the number of errors that the code surely detects and the probability of undetected errors.