The decoding of Reed-Solomon codes

Reed-Solomon (RS) codes form an important part of the high-rate downlink telemetry system for the Magellan mission, and the RS decoding function for this project will be done by DSN. Although the basic idea behind all Reed-Solomon decoding algorithms was developed by Berlekamp in 1968, there are dozens of variants of Berlekamp's algorithm in current use. An attempt to restore order is made by presenting a mathematical theory which explains the working of almost all known RS decoding algorithms. The key innovation that makes this possible is the unified approach to the solution of the key equation, which simultaneously describes the Berlekamp, Berlekamp-Massey, Euclid, and continued fractions approaches. Additionally, a detailed analysis is made of what can happen to a generic RS decoding algorithm when the number of errors and erasures exceeds the code's designed correction capability, and it is shown that while most published algorithms do not detect as many of these error-erasure patterns as possible, by making a small change in the algorithms, this problem can be overcome.