Error-Detecting Code Using Linear Quasigroups

Abstract

In this paper we consider an error-detecting code based on linear quasigroups of order 2^q defined in the following way: The input block a ₀ a ₁...a _n − 1 is extended into a block a ₀ a ₁...a _n − 1 d ₀ d ₁...d _n − 1, where redundant characters d ₀ d ₁...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 the probability of undetected errors is independent from the distribution of the characters in the input message. We also calculate the probability of undetected errors, if quasigroups of order 8 are used. We found a class of quasigroups of order 8 that have smallest probability of undetected errors, i.e. the quasigroups which are the best for coding. We explain how the probability of undetected errors can be made arbitrary small.