Abstract
A theoretical representation of the CRC codes with the aid of the finite automata in the Galois fields is proposed –linear finite-state machine (LFSM). We examined three types of LSC, which are distinguished by its characteristic matrices: type 1, type 2, and type 3. Three methods of the parallel CRC computation are proposed. The theorems are proven about the estimation of duration of the CRC computation by tuple-parallel method for three types of LFSM. We presented algorithms for the CRC tuple-parallel computation by the tables lookup for different types of LFSM and demonstrated their hardware implementation. For the symbolic-parallel and symbolic-tuple-parallel methods of CRC computation, symbolic LFSM over the non-binary Galois fields are applied. In contrast to the known approaches, which consider the problem of high speed only, present article stresses the need for taking into account a capability to detect errors with the aid of generator polynomials of the CRC codes. We proposed the Hamming codes and the Abramson codes in the binary and non-binary Galois fields for this purpose. Within the framework of single mathematical apparatus – theory of LFSM – it is possible to solve two important tasks simultaneously: to carry out mathematical substantiation of performance efficiency of the methods for parallel CRC computation and to estimate the detection and correction capability of the CRC codes based on LFSM graph models. Such comprehensive approach is relevant in the practice of various data transmission, storage and compression systems.
Highlights
CRC-check belongs to the most commonly used methods of check in different information systems
If a symbol’s bit capacity is equal to length h of tuple at the tuple-parallel method of CRC computation, the duration of time T3 will be nearly equal to the duration of time T2. (Let us note that here we are abstracted from the details of technical implementation and we examine only mathematical time)
We examined a method for the CRC codes representation on the basis of mathematical apparatus of linear finite-state machine (LFSM) and conducted comparative theoretical analysis of the LFSM properties of three types
Summary
CRC-check belongs to the most commonly used methods of check in different information systems. CRC is the method for detecting errors, which has very simple software and hardware implementation. This advantage manifests itself only at the serial entering of input data, which was characteristic in the early years of development of information systems. In contemporary high-speed data transmission, especially in the multichannel communication systems, large delays occur in the realization of procedures of searching for errors with the aid of traditional CRC. That is why it is absolutely relevant to simultaneously solve the problems on the provision of high performance speed of CRC codes and on the detection of the maximum number of various data distortions by them
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