Polynomial Codes Properties Application in Concurrent Error-Detection Systems of Combinational Logic Devices
Аннотация
This paper presents the revealed patterns of constructing polynomial codes with different detecting characteristics. The dependence of the number of the code check groups over which the data vectors are distributed on the form of the generating polynomial is established. It is proved that only in the presence of a free term in the generating polynomial, the maximum possible number of check groups of the code is used for a given length of the check vector. A formula is obtained for calculating the number of undetectable errors by polynomial codes constructed using various generators of polynomials. A theorem is proved about the presence among polynomial codes of codes with a theoretical indicator of the minimum number of undetectable errors for given values of the lengths of data and check vectors. As the length of the check vector increases, the number of such codes also increases. A theorem is proved on the existence of parity codes with better detecting characteristics in comparison with classical parity codes. The useful properties of the considered codes for the synthesis tasks of systems with error detection are presented. In the example of benchmarks, experiments were carried out to evaluate the detecting characteristics of special polynomial codes. For some polynomials and benchmarks, complete error-detection at their outputs has been achieved. Besides, the occupied areas of concurrent error-detection systems based on the considered codes and the duplication method were calculated and compared, according to the results of which, even with such a large value of the check vector length, systems based on polynomial codes have a significant advantage in hardware costs.