Abstract
The object of research is the processes of error-correcting coding in telecommunication and computer systems. The main attention is paid to Reed-Solomon (RS) codes, which belong to the very widespread error-correcting codes. Despite the 60-year existence of these codes, the complexity of their decoding still remains a problem. This problem is mainly due to the use of an algebraic approach to their description. The article proposes to use the theory of linear finite-state machine (LFSM) for RS codes as a mathematical basis, which is a combination of the theory of digital filters and finite automaton over nonbinary Galois fields. In the course of research, 12 types of LFSMs are considered for the first time: the recursive LFSMs of 8 types and the non-recursive LFSMs of 4 types. The recursive LFSMs are used for systematic encoding and form a circuit for dividing of polynomials, and the non-recursive LFSMs are used for non-systematic encoding and form a circuit for multiplying of polynomials. All types of LFSMs give the same result for encoding and decoding, but with different complexity, which is impor-tant for practical implementation. The automaton representation is the most suitable for RS codes, since it takes into account the cyclicity property and other features of these codes to the maximum. In contrast to algebraic methods, automaton decoding methods have a simple software and hardware implementation and high performance. With the help of automaton-graphical models, it can accurately estimate the corrective capability of the code. Automaton representation combines known methods of representing Reed-Solomon codes (polynomial, matrix, algebraic) and provides mutual transitions between them. The article attention is spare to the fact that automaton methods for encoding and decoding (n, k)-codes of RS using quantum computers give a gain in time n times.
Highlights
This paper is the extended version of conference paper [1]
The automaton representation is the most suitable for RS codes, since it takes into account the cyclicity property and other features of these codes to the maximum
A k -symbol information word is fed to the input of the recursive linear finite-state machine (LFSM), as a result of which the LFSM will move from the initial zero state S(0) to the state S(k) during the k time clocks according to the formula that follows from (2): S(k) = Ak × S(0) + Lk × I, GF (q), The state S(n) is the error syndrome: a zero value of this state indicates the absence of errors in the transmitted codeword within the corrective capacity of the code
Summary
The Reed-Solomon (RS) codes appeared 60 years ago and until now they have included into the best errorcorrecting codes. New codes and new principles for decoding error-correcting codes began to appear, but in RS codes, as in other subclasses of cyclic codes, the laborious and inconvenient Berlekamp-Massey method remains the main decoding method [3]. This situation has developed mainly because cyclic codes are considered only as a subclass of linear codes and, universal algebraic methods for decoding linear codes are try adapted to these codes. The aim of research is to improve the efficiency of means of transmission, storage, processing and protection of data through the development of new theoretical models of RS codes
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