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]. |

Classical Goppa code | The McEliece public-key cryptosystem [7,8]. 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 \(n\) [9]. |

Covering code | Data compression both with or without compression [10].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 [11,12]. |

Cycle LDPC code | Cycle LDPC codes have been proposed to be used for MIMO channels [13]. |

Cyclic redundancy check (CRC) code | CRC-16 and CRC-32 are used in data transmission, e.g., IEEE 802.16e, IEEE 802.3 [14] and TCP/IP communication [15; Sec. 2.3.3]. |

Delsarte-Goethals (DG) code | Space-time signaling [16].Compressed neighbor discovery in a network [17]. |

Fountain code | Designed for servers sending data to many recipients, such as during broadcasting or file distribution [18,19].DNA storage [20]. |

Gabidulin code | Public-key cryptosystems [21,22].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 [23]. |

Generalized RS (GRS) code | Commonly used in mass storage systems such as CDs, DVDs, QR codes etc.Various cloud storage systems [24].A variation of the McEliece public-key cryptosystem [7,8] by Niederreiter [25] 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 [26]. More recent works focus on methods to mask the algebraic structure using subcodes of GRS codes [21]. For example, a key-recovery attack was developed in Ref. [27] for a variant of masking method proposed in Ref. [28]. |

Golay code | Extended Golay code used in the Voyager 1 and 2 spacecraft, transmitting hundreds of color pictures of Jupiter and Saturn in their 1979, 1980, and 1981 fly-bys [29].Extended Golay code used in American military standards for automatic link establishment in high frequency radio systems [30].Proofs of the quantum mechanical Kochen-Specker theorem [31]. |

Gold code | Used in for synchronization purposes in telecommunication [32]GPS C/A for satellite navigation [33]. |

Gray code | Three-dimensional imaging [34].Broadcasting and communication [35]. |

Hessian polyhedron code | Quantum mechanical SIC-POVMs [36]. |

Interleaved RS (IRS) code | The cross-interleaved RS (CIRC), an IRS code using two shortened RS codes and two forms of interleaving, was used for compact discs (CDs) [37] (see Ref. [38], Sec. 5.6 and Ref. [39], Ch. 4). |

Irregular LDPC code | Satellite communication after concatenating with a modulation scheme [40]. |

Irregular repeat-accumulate (IRA) code | LDPC codes are used for digital satellite video broadcasting per the DVB-S2 standard [41,42] utilize IRA code features and are subject to ongoing litigation; see Ref. [43].Apple and Broadcom Wi-Fi devices utilize IRA encoding and decoding code features and are subject to ongoing litigation; see Ref. [43]. |

Justesen code | Generating small-bias sample spaces, i.e., probability distributions that parity functions cannot typically distinguish from the uniform distribution [44]. |

Low-rank parity-check (LRPC) code | Cryptosystem [45] that is a rank-metric analogue of NTRU [46] and MDPC [47] cryptosystems.Post-quantum cryptography [48]. |

Maximum distance separable (MDS) code | Automatic repeat request (ARQ) data transmission protocols ([39], Ch. 7). |

Maximum-rank distance (MRD) code | Useful for error and erasure correction in network coding [49,50]. |

Multi-edge LDPC code | Quantum key distribution [51–53]. |

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 ([54], Ch. 27).Marking the state of a finite automaton [55]. |

Phase-shift keying (PSK) code | Telephone-line modems: 1967 Milgo 4400/48 and international standard V.27 used 8-PSK [56]. |

Polar code | Code control channels for the 5G NR (New Radio) interfaces [57]. |

Pulse-position modulation (PPM) code | Greek hydraulic semaphore system [58,59].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 [60].Telephone-line modems: 1962 Bell 201 and international standard V.24 [61]. |

Quadrature-amplitude modulation (QAM) code | Optical communication (e.g., Ref. [62]).Telephone-line modems: 1971 Codex 9600C and international standard V.29 used 16-QAM [63]. |

Quasi-cyclic LDPC (QC-LDPC) code | 5G NR cellular communication for the traffic channel [64,65].Wireless communication: WiMAX (IEEE 802.16e) [66–68], WiFi 4 (IEEE 802.11n) [69], and WPAN (IEEE 802.15.3c) [70]. |

Rank-metric code | Identity-Based Encryption [71].Digital watermarking [72].Network coding and streaming media broadcasting [73]. |

Rank-modulation Gray code (RMGC) | Electronic devices where charges can either increase in an individual cell or decrease in a block of adjacent cells, e.g., flash memories [74]. |

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 [75,76]. |

Reed-Solomon (RS) code | RS Product Code (RSPC) was used in DVDs (see Ref. [39], Ch. 4).DSL technologies and their variants against impluse noise [77].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 [78].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. [39], Ch. 3).Automatic repeat request (ARQ) data transmission protocols (see Ref. [39], Ch. 7).Slow-frequency-hop spread-spectrum transmission (see Ref. [39], Chs. 8-9).Coded sharding designs in blockchains to increase efficiency [79].Private Information Retrieval [80].Used in QR-Codes to retrieve damaged barcodes [81].Wireless communication systems such as 3G, DVB, and WiMAX [82].Correcting pooled testing results for SARS-CoV-2 [83].DNA storage [84]. |

Regular binary Tanner code | First hardware implementation was done using a semi-systolic decoding architecture [85]. |

Repetition code | Repetition codes, in conjunction with other codes, were used in magnetic disks [86].Communication protocols such as FlexRay [87].' |

Residue AG code | Improvements over the McEliece public-key cryptosystem to linear AG codes on curves of arbitrary genus [88]. Only the subfield subcode proposal remains resilient to attacks [89; Sec. 15.7.5.3].Algebraic geometric secret-sharing schemes [90]. |

Single parity-check (SPC) code | Can be realized on almost every communication device. SPCs are some of the earliest error-correcting codes ([54], Ch. 27). |

Spacetime block code (STBC) | High data-rate wireless communication, e.g., WiMAX (IEEE 802.16m) [91–93]. |

Tensor-product code | Construction can be used in magnetic recording by taking the tensor product of an RS code and a parity-check code [94]. |

Ternary Golay code | Code used in football pools with at least one good bet [12,95]. In fact, the code was originally constructed by Juhani Virtakallio and published in the Finnish football pool magazine Veikkaaja [12,96].Proofs of the quantum mechanical Kochen-Specker theorem [31]. |

Unary code | Neural networks [97].Birdsong production [98]. |

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 [96,99]. |

Zetterberg code | Code used to provide better protection of data transmission with its double error correcting capacity [100]. |

\(A_2\) hexagonal lattice code | Wireless communication [101,102]. |

\([2^m,m+1,2^{m-1}]\) First-order RM code | The \([32, 6, 16]\) RM\((1,5)\) code was used for the 1971 Mariner 9 spacecraft [103]. |

\([2^r-1,2^r-r-1,3]\) Hamming code | Commonly used when error rates are very low, for example, computer RAM or integrated circuits [104].Hamming-code based matrix embedding used in steganography [105,106]. |

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