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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.
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 ...
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.
The most popular erasure codes are Reed-Solomon coding, Low-density parity-check code (LDPC codes), and Turbo codes. [ 1 ] As of 2023, modern data storage systems can be designed to tolerate the complete failure of a few disks without data loss, using one of 3 approaches: [ 2 ] [ 3 ] [ 4 ]
It is used as one of the steps in decoding BCH codes and Reed–Solomon codes ... (PDF), Stanford University, pp. 42– 45, archived from the original (PDF) on June ...
Practical implementations rely heavily on decoding the constituent SPC codes in parallel. LDPC codes were first introduced by Robert G. Gallager in his PhD thesis in 1960, but due to the computational effort in implementing encoder and decoder and the introduction of Reed–Solomon codes, they were mostly ignored until the 1990s.
Consider a (,) Reed–Solomon code over the finite field = with evaluation set (,, …,) and a positive integer , the Guruswami-Sudan List Decoder accepts a vector = (,, …,) as input, and outputs a list of polynomials of degree which are in 1 to 1 correspondence with codewords.
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.