A survey of constructive coding theory, and a table of binary codes of highest known rate (original) (raw)

Abstract

Although n-me than f wenty years have passed since the appearance of Shannon's iapers, a stilll unsolved problem o: coding theory is to construct block codes which attain a low probability of error at rates clax to capacity. However, for moderate block lengths many good codes are known!, the best-known being the BCH codes discovered in 1959. This paper is a survey of results in coding theory obtainedsisce the appearance of Berlekamp's'"Algebraic coding theory" (1968), concentrating on those which lead. to the construction of new codes. The paper concludes with ir table giving the smallest redundancy of any binary code, linear or nonlinear, that is presently known (to the author), for all lengths up to 5 12 and all minimum distances up to 30.

FAQs

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What are the key properties of cyclic and non-linear codes?add

The paper highlights that cyclic codes are simpler and generally yield better performance metrics compared to non-linear codes. For instance, cyclic codes like BCH show minimum distances that meet Gilbert bounds.

How did BCH codes improve error-correction capabilities?add

BCH codes achieve significant error-correction capability by utilizing prime powers, exhibiting minimum distances of at least 2t + 1. These codes are constructed to meet the Gilbert boundary for long code lengths.

What metrics are used to evaluate binary codes in this context?add

The evaluation of binary codes primarily considers parameters like redundancy, minimum distance, and the number of codewords. The redundancy is calculated as r = n - log_q(M), connecting code length and dimensions.

What does recent research suggest about self-dual linear codes?add

Recent findings indicate that self-dual linear binary codes with weights divisible by 4 meet the Gilbert bound, shedding light on optimal characteristics. For example, Thompson demonstrated their efficacy in specific coding scenarios.

When were the earliest findings on error-correcting codes established?add

The foundational work in error correction dates back to 1948 when Shannon predicted codes could achieve error probabilities close to channel capacity. Gilbert later established lower bounds on code performance metrics in the following decades.

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