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The QR code, Ver 3 (29×29) uses interleaved blocks. The message has 26 data bytes and is encoded using two Reed-Solomon code blocks. Each block is a (255,233) Reed Solomon code shortened to a (35,13) code. The Delsarte–Goethals–Seidel [12] theorem illustrates an example of an application of shortened Reed–Solomon codes.
Block codes are processed on a block-by-block basis. Early examples of block codes are repetition codes, Hamming codes and multidimensional parity-check codes. They were followed by a number of efficient codes, Reed–Solomon codes being the most notable due to their current widespread use.
This process is implemented by Reed–Solomon codes, with code words constructed over a finite field using a Vandermonde matrix. Most practical erasure codes are systematic codes -- each one of the original k symbols can be found copied, unencoded, as one of the n message symbols. [ 12 ] (
The term "folded Reed–Solomon codes" was coined in a paper by V.Y. Krachkovsky with an algorithm that presented Reed–Solomon codes with many random "phased burst" errors. [1] The list-decoding algorithm for folded RS codes corrects beyond the 1 − R {\displaystyle 1-{\sqrt {R}}} bound for Reed–Solomon codes achieved by the Guruswami ...
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English: This is a pictorial representation of a code concatenation, and, in particular, the Reed–Solomon code with n=q=4 and k=2 is used as the outer code and the Hadamard code with n=q and k=log q is used as the inner code. Overall, the concatenated code is a [, ]-code.
Algebraic geometry codes are a generalization of Reed–Solomon codes. Constructed by Irving Reed and Gustave Solomon in 1960, Reed–Solomon codes use univariate polynomials to form codewords, by evaluating polynomials of sufficiently small degree at the points in a finite field. [8] Formally, Reed–Solomon codes are defined in the following way.
The codes that they are given are called folded Reed-Solomon codes which are nothing but plain Reed-Solomon codes but viewed as a code over a larger alphabet by careful bundling of codeword symbols. Because of their ubiquity and the nice algebraic properties they possess, list-decoding algorithms for Reed–Solomon codes were a main focus of ...