The Error Correction Zoo

Victor V. Albert; Philippe Faist

Here is a list of classical codes that have been realized in devices and/or that have practical real-world relevance.

Name	Realization(s)
Alamouti code	Wireless standards since: 3G, LTE, LTE-Advanced, and 5G.Wireless communication: IEEE 802.11n, IEEE 802.11ad, IEEE 802.11ay, etc.
Array-based LDPC (AB-LDPC) code	Certain AB-LDPC codes have been proposed to be used for DSL transmission [1].
Balanced code	Balanced length-eight code, known as a 6b/8b encoding, used for balancing direct current in a communications system [2]
Binary BCH code	Satellite communication [3]
Binary PSK (BPSK) code	Telephone-line modems throughout 1950s and 1960s: Bell 103 and 202, as well as international standards V.21 [4] and V.23 [5].
Bose–Chaudhuri–Hocquenghem (BCH) code	DVDs, disk drives, and two-dimensional bar codes [6].
Constant-weight code	Radio-network frequency hopping [7].
Convolutional code	A type of convolutional code used in Real-time Application networks [8].Mobile and radio communications (3G networks) use convolutional codes concatenated with RS codes to obtain suitable performance [9].A convolutional code with rate 1/2 was used for deep-space and satellite communication [10]
Covering code	Data compression both with or without compression [11].Football-pool problem: finding the smallest number of bets on a set of matches needed to guarantee at least one bet has at most \(\rho\) errors [12,13].
Cross-interleaved RS (CIRS) code	Compact discs (CDs); see [15][14; Sec. 5.6][16; Ch. 4].
Cycle LDPC code	Cycle LDPC codes have been proposed to be used for MIMO channels [17].
Cyclic redundancy check (CRC) code	CRC-16 and CRC-32 are used in data transmission, e.g., IEEE 802.16e, IEEE 802.3 [18] and TCP/IP communication [19; Sec. 2.3.3].
Delsarte-Goethals (DG) code	Space-time signaling [20].Compressed neighbor discovery in a network [21].
Error-correcting output code (ECOC)	Multiclass problems in machine learning, relevant to facial recognition [22,23], text recognition [24], or digit classification [25].
Fountain code	Designed for servers sending data to many recipients, such as during broadcasting or file distribution [26,27].DNA storage [28].
Frameproof (FP) code	FP codes are utilized in digital fingerprinting and watermarking [29].
Gabidulin code	Public-key cryptosystems [30,31].Digital watermarking. The Gabidulin code provides efficient correction against luminance tampering and image-slicing distortion due to the consistency of the rank against alterations such as column swapping [32].
Generalized RS (GRS) code	Commonly used in mass storage systems such as CDs, DVDs, QR codes etc.Various cloud storage systems [33].A variation of the McEliece public-key cryptosystem [34,35] by Niederreiter [36] replaced the generator matrix by the parity check matrix of a GRS code. This was proven to be insecure since the public key exposes the algebraic structure of code [37]. More recent works focus on methods to mask the algebraic structure using subcodes of GRS codes [30]. For example, a key-recovery attack was developed in Ref. [38] for a variant of masking method proposed in Ref. [39].
Gold code	Used in for synchronization purposes in telecommunication [40]GPS C/A for satellite navigation [41].
Goppa code	The McEliece public-key cryptosystem [34,35]. The protocol relies on the assumptions that Goppa-code generator matrices are hard to distinguish from random linear codes. However, there is an algorithm distinguishing between the two code classes in a time subexponential in \(n\) [42].
Gray code	Three-dimensional imaging [43].Broadcasting and communication [44].
Hessian polyhedron code	Quantum mechanical SIC-POVMs [45].
Irregular LDPC code	Satellite communication after concatenating with a modulation scheme [46].
Irregular repeat-accumulate (IRA) code	LDPC codes are used for digital satellite video broadcasting per the DVB-S2 standard [47,48] utilize IRA code features and are subject to ongoing litigation; see Ref. [49].Apple and Broadcom Wi-Fi devices utilize IRA encoding and decoding code features and are subject to ongoing litigation; see Ref. [49].
Justesen code	Generating small-bias sample spaces, i.e., probability distributions that parity functions cannot typically distinguish from the uniform distribution [50].
Linearized RS code	Network coding [51].Code-based cryptography [52,53].
Locally recoverable code (LRC)	An \((18,14,7)\) LRC has beed used in the Windows Azure cloud storage system [54]; see also [55; 31.3.1.2].Facebook f4 BLOB cloud storage system [56]
Low-rank parity-check (LRPC) code	Cryptosystem [57] that is a rank-metric analogue of NTRU [58] and MDPC [59] cryptosystems.Post-quantum cryptography [60].
Maximally recoverable (MR) code	RAID storage [61].
Maximum distance separable (MDS) code	Automatic repeat request (ARQ) data transmission protocols ([16], Ch. 7).
Maximum-rank distance (MRD) code	Useful for error and erasure correction in network coding [62,63].
Multi-edge LDPC code	Quantum key distribution [64–66].
One-hot code	The bi-quinary code, a combination of one-hot 1-in-2 and 1-in-5 one-hot codes to encode decimal digits, was used in several early computers ([67], Ch. 27).Marking the state of a finite automaton [68].Encoding the output of the different classes of a classifier neural network [69]. One-hot codes are the primary codes used in multiclass classification [70–73].
Phase-shift keying (PSK) code	Telephone-line modems: 1967 Milgo 4400/48 and international standard V.27 used 8-PSK [74].
Polar code	Code control channels for the 5G NR (New Radio) interfaces [75].
Pulse-position modulation (PPM) code	Greek hydraulic semaphore system [76,77].Telegraph time-division multiplexing.Radio-control, fiber-optic communications, and deep-space communications.
Quadrature PSK (QPSK) code	Japanese and North American digital cellular and personal systems [78].Telephone-line modems: 1962 Bell 201 and international standard V.24 [79].
Quadrature-amplitude modulation (QAM) code	Optical communication (e.g., Ref. [80]).Telephone-line modems: 1971 Codex 9600C and international standard V.29 used 16-QAM [81].
Quasi-cyclic LDPC (QC-LDPC) code	5G NR cellular communication for the traffic channel [82,83].Wireless communication: WiMAX (IEEE 802.16e) [84–86], WiFi 4 (IEEE 802.11n) [87], and WPAN (IEEE 802.15.3c) [88].
Random code	Distributed storage systems [89].Classical and quantum cryptography based on the learning-with-errors problem, which is related to decoding a random linear code [90].Random codes can be used to realize secure computation [91].
Rank-metric code	Identity-Based Encryption [92].Digital watermarking [93].Network coding and streaming media broadcasting [94].
Rank-modulation code	Electronic devices where charges can either increase in an individual cell or decrease in a block of adjacent cells, e.g., flash memories [95].
Raptor (RAPid TORnado) code	Two versions of Raptor codes have been standardized by IETF: R10 and the more recent RaptorQ. RaptorQ is used in mobile multimedia broadcasts as specified in ETSI technical specifications. It is also used in the mobile Next Gen TV standard.Raptor codes are useful in scenarios where erasure (i.e. weak signal or noisy channel) is common, such as in military or disaster scenarios.
Reed-Muller (RM) code	Deep-space communication [96,97].
Reed-Solomon (RS) code	RS Product Code (RSPC) was used in DVDs (see Ref. [16], Ch. 4).DSL technologies and their variants against impluse noise [98].Cryptographic primitives based on the hardness of decoding RS codes for more than \(1-\sqrt{k/n}+\epsilon\) errors. This is equivalent to the polynomial reconstruction problem [99].RS codes as outer codes concatenated with convolutional codes are used indirectly in space exploration programs such as Voyager and Galileo. RS codes were part of a telemetry channel coding standard issued by the Consultative Committee for Space Data Systems (see Ref. [16], Ch. 3).Automatic repeat request (ARQ) data transmission protocols (see Ref. [16], Ch. 7).Slow-frequency-hop spread-spectrum transmission (see Ref. [16], Chs. 8-9).RS codes over \(q=2^m\) are used in RAID 6 [100,101]; see [15].Coded sharding designs in blockchains to increase efficiency [102].Used in QR-Codes to retrieve damaged barcodes [103].Wireless communication systems such as 3G, DVB, and WiMAX [104].Correcting pooled testing results for SARS-CoV-2 [105].DNA storage [106].
Regular binary Tanner code	First hardware implementation was done using a semi-systolic decoding architecture [107].
Repetition code	Repetition codes, in conjunction with other codes, were used in magnetic disks [108].Communication protocols such as FlexRay [109].'
Residue AG code	Improvements over the McEliece public-key cryptosystem to linear AG codes on curves of arbitrary genus [110]. Only the subfield subcode proposal remains resilient to attacks [111; Sec. 15.7.5.3].Algebraic geometric secret-sharing schemes [112].
Single parity-check (SPC) code	Can be realized on almost every communication device. SPCs are some of the earliest error-correcting codes ([67], Ch. 27).
Skew-cyclic code	Not directly implemented, but BCH codes form a subclass, and are used in DVD, solid state drive storage, etc.
Spacetime block code (STBC)	High data-rate wireless communication, e.g., WiMAX (IEEE 802.16m) [113–115].
Spherical code	Spherical codes are relevant to modern Hopfield networks [116,117]
Subspace code	Packet-based transmission over networks [118].
Tensor-product code	Construction can be used in magnetic recording by taking the tensor product of an RS code and a parity-check code [119].
Ternary Golay code	Code used in football pools with at least one good bet [13,120]. In fact, the code was originally constructed by Juhani Virtakallio and published in the Finnish football pool magazine Veikkaaja [13,121,122].Proofs of the quantum mechanical Kochen-Specker theorem [123].
Traceability code	Broadcast messages, pay-per-view movies, and protecting copyrighted online material [124].
Turbo code	Recommended by Consultative Committee on Space Data Systems (CCDJS) for telemetry and telecommand [125,126].Several standards related to wireless communication, including W-CDMA, DVB-RCS, TD-SCDMA, 802.16, and CDMA2000 [127].
Unary code	Neural networks [128].Birdsong production [129].
Weight-two code	Two-in-five, also known as the two-out-of-five code, was used in the United States Postal Service's POSTNET barcode system as well as the Postal Alpha-numeric Encoding Technique (PLANET).Two-in-five code forms the numerical part of the Code 39 barcode encoding.Two-in-five code was used on early IBM computers [122,130].
Zetterberg code	Code used to provide better protection of data transmission with its double error correcting capacity [131].
\(A_2\) triangular lattice	Wireless communication [132,133].
\([23, 12, 7]\) Golay code	Proofs of the quantum mechanical Kochen-Specker theorem [123].
\([24, 12, 8]\) Extended Golay code	Voyager 1 and 2 spacecraft, transmitting hundreds of color pictures of Jupiter and Saturn in their 1979, 1980, and 1981 fly-bys [134].American military standards for automatic link establishment in high frequency radio systems [135].
\([2^m,m+1,2^{m-1}]\) First-order RM code	The \([32, 6, 16]\) RM\((1,5)\) code was used for the 1971 Mariner 9 spacecraft [136].
\([2^r-1,2^r-r-1,3]\) Hamming code	Commonly used when error rates are very low, for example, computer RAM or integrated circuits [137].Hamming-code based matrix embedding used in steganography [138,139].

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