Calculation of the detection properties in the binary symmetrical channel

Abstract

In safety critical applications (e.g. in interlocking systems) it is essential to exactly quantify the probability of hazard events. Coding theory is one of the fields connected with the safety problems. Especially block detection codes are basic method for data integrity check. The requirements on the properties of detection codes are specified in the EU norm EN 50159-1[1]. One of the most important safety parameters of the code is the probability of the undetected error in the binary symmetrical channel. To compute the probability of the undetected error it is necessary to know the weight distribution of the code. This problem is quite time consuming, especially for long code words. Therefore the European norm suggests using an estimate of the maximum probability of the undetected error that is based only on the number of redundant bytes. In the article a method of computing the weight distribution of the code with the help of the dual code using MacWilliams formula is demonstrated. The weight distributions for all binary primitive polynomials of the order 16 and 8 are considered. It appears that the concept of proving safety using the worst case is not fulfilled by the estimation suggested in the mentioned norm. In consequence, the recommendation of the norm to use this estimate for safety critical applications cannot be respected and the computation of the weight distribution has to be performed. Computational difficulty of the calculation of the weight distribution of the code is then discussed. The detection ability of the primitive polynomials is investigated.