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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.
Reed-Solomon codes are used in compact discs to correct errors caused by scratches. Modern hard drives use Reed–Solomon codes to detect and correct minor errors in sector reads, and to recover corrupted data from failing sectors and store that data in the spare sectors. [20]
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 ]
For example: The Reed-Solomon code with LDPC Coded Modulation (RS-LCM) uses a Reed-Solomon outer code. [18] The DVB-S2, the DVB-T2 and the DVB-C2 standards all use a BCH code outer code to mop up residual errors after LDPC decoding. [19] 5G NR uses polar code for the control channels and LDPC for the data channels. [20] [21]
The combination of an inner Viterbi convolutional code with an outer Reed–Solomon code (known as an RSV code) was first used in Voyager 2, [5] [8] and it became a popular construction both within and outside of the space sector.
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.
This page was last edited on 26 January 2025, at 03:50 (UTC).; Text is available under the Creative Commons Attribution-ShareAlike 4.0 License; additional terms may apply.
The BCH code over () and generator polynomial () with successive powers of as roots is one type of Reed–Solomon code where the decoder (syndromes) alphabet is the same as the channel (data and generator polynomial) alphabet, all elements of (). [6]