Springer papers

### Authors

Natasha
Ilievska
and
Danilo
Gligoroski
### 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 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.

### Keywords

error-detecting codes linear quasigroup noisy channel binary symmetric channel probability of undetected errors