The Error Correction Zoo

Victor V. Albert; Philippe Faist

Here is a list of all classical and quantum codes that have been realized in experimental or real-world devices.

Name	Realization(s)
120-cell code	Improved proofs of the Bell-Kochen-Specker (BKS) theorem [1].
600-cell code	Improved proofs of the Bell-Kochen-Specker (BKS) theorem [2].
Alamouti code	Wireless standards since: 3G, LTE, LTE-Advanced, and 5G.Wireless communication: IEEE 802.11n, IEEE 802.11ad, IEEE 802.11ay, etc.
Analog stabilizer code	One-sided device-independent QKD [3].
Array-based LDPC (AB-LDPC) code	Certain AB-LDPC codes have been proposed to be used for DSL transmission [4].
BPSK c-q code	Linear-optical quantum receiver [5].Homodyne receiver [6].Kennedy receiver [6,7].Photon-number resolving detector [8].Communication over dephasing [9], time-varying phase-noise [10], and thermal-noise [11] channels.Adaptive decoder using displacements and photon detection [12].Superconducting circuits: a bit-flip time of 1s has been achieved for the two-legged cat code in the classical-bit regime [13].
Bacon-Shor code	Trapped-ion qubits: state preparation, logical measurement, and stabilizer measurement for nine-qubit Bacon-Shor code demonstrated on a 13-qubit device by M. Cetina and C. Monroe groups [14].
Balanced code	Balanced length-eight code, known as a 6b/8b encoding, used for balancing direct current in a communications system [15]
Binary BCH code	Satellite communication [16]
Binary PSK (BPSK) code	Telephone-line modems throughout 1950s and 1960s: Bell 103 and 202, as well as international standards V.21 [17] and V.23 [18].
Binomial code	Microwave cavities coupled to superconducting circuits: state transfer between a binomial codeword to another system [19], error-correction protocol nearly reaching break-even [20], and a teleported CNOT gate [21]. A realization of the "0-2-4" encoding is the first to go beyond break-even error-correction and yields a logical lifetime that exceeds the cavity lifetime by $16\%$ [22] (see also [23]). See Ref. [24] for another experiment.Motional degree of freedom of a trapped ion: binomial state preparation for $S=2$ realized by Tan group [25].
Bose–Chaudhuri–Hocquenghem (BCH) code	DVDs, disk drives, and two-dimensional bar codes [26].
Cat code	Parity-syndrome measurement tested [27] and implemented for the four-component ($S=1$) cat code [28] in a microwave cavity coupled to a superconducting circuit. The latter work [28] is the first to reach break-even error-correction, where the lifetime of a logical qubit is on par with the cavity lifetime, despite protection against dephasing not being implemented. A fault-tolerant version of parity measurement has also been realized [29].
Classical Goppa code	Initial version of the McEliece public-key cryptosystem [30,31] and its variation by Niederreiter [32] where the generator matrix is replaced by the parity check matrix. Some of these were proven to be insecure since the public key exposes algebraic structure of code [33].
Coherent FSK (CFSK) c-q code	Time-resolving quantum receiver [34].Bondurant receiver [35].Bayesian inference [36].
Coherent-state c-q code	Continuous-variable quantum key distribution (CV-QKD) [37–39].
Constant-weight code	Radio-network frequency hopping [40].
Convolutional code	A type of convolutional code used in Real-time Application networks [41].Mobile and radio communications (3G networks) use convolutional codes concatenated with Reed-Solomon codes to obtain suitable performance [42].A convolutional code with rate 1/2 was used for deep-space and satellite communication [43]
Covering code	Data compression both with or without compression [44].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 [45,46].
Cycle LDPC code	Cycle LDPC codes have been proposed to be used for MIMO channels [47].
Dihedral $G=D_m$ quantum-double code	Signatures of a phase equivalent to the $G=D_4$ quantum double detected in a 27-qubit trapped-ion device by Quantinuum [48]. Preparation of ground states and braiding of anyons has also been performed. The phase was realized as a gauged $G=\mathbb{Z}_3^2$ twisted quantum double [49], which is the same topological order as the $G=D_4$ quantum double [50,51].
Dual-rail quantum code	The dual-rail code is ubiquitous in linear optical quantum devices [52].Superconducting circuit devices: error detection has been demonstrated in 3D cavities in Michel Devoret's group at Yale University [53] and Amazon Web Services [54] using transmon qubits, following earlier theoretical proposals [55,56]. Logical readout in 3D cavities has been demonstrated by Quantum Circuits Inc. [57].
Fibonacci string-net code	NMR: Implementation of braiding-based Hamamard gate on two qubits [58].
Five-qubit perfect code	NMR: Implementation of perfect error correcting code on 5 spin subsystem of labeled crotonic acid for quantum network benchmarking [59]. Single-qubit logical gates [60]. Magic-state distillation using 7-qubit device [61].Superconducting qubits [62].Trapped-ion qubits: non-transversal CNOT gate between two logical qubits, including rounds of correction and fault-tolerant primitives such as flag qubits and pieceable fault tolerance, on a 12-qubit device by Quantinuum [63]. Real-time magic-state distillation [64].Nitrogen-vacancy centers in diamond: fault-tolerant single-qubit Clifford operations [65].
Floquet color code	Plaquette stabilizer measurement realized on the IBM Falcon superconducting-qubit device [66]
Fountain code	Designed for servers sending data to many recipients, such as during broadcasting or file distribution [67,68].
Gabidulin code	Public-key cryptosystems [69,70].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 [71].
Generalized RS (GRS) code	Commonly used in mass storage systems such as CDs, DVDs, QR codes etc.Various cloud storage systems [72].Public-key cryptosystems generalizing those that used Goppa codes [30–32], some of which were proven to be insecure [33]. More recent works focus on methods to mask the algebraic structure using subcodes of GRS codes [69]. For example, a key-recovery attack was developed in Ref. [73] for a variant of masking method proposed in Ref. [74].
Generalized Shor code	The $[[m^2,1,m]]$ codes for $m\leq 7$ have been realized in trapped-ion quantum devices [75].
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 [76].Extended Golay code used in American military standards for automatic link establishment in high frequency radio systems [77].Proofs of the quantum mechanical Kochen-Specker theorem [78].
Gold code	Used in for synchronization purposes in telecommunication [79]GPS C/A for satellite navigation [80].
Gray code	Three-dimensional imaging [81].Broadcasting and communication [82].
Group GKP code	Cryptographic applications stemming from the monogamy of entanglement of group GKP code and error words [3].
Hamming code	Commonly used when error rates are very low, for example, computer RAM or integrated circuits [83].Hamming-code based matrix embedding used in steganography [84,85].
Heavy-hexagon code	Logical state preparation and flag-qubit error correction realized in superconducting-circuit devices (specifically, fixed-frequency transmon qubit architectures) by IBM for $d=2$ [86,87] and $d=3$ [88].
Hessian polyhedron code	Quantum mechanical SIC-POVMs [89].
Hexagonal GKP code	Microwave cavity coupled to superconducting circuits: reduced form of GKP error correction, where displacement error syndromes are measured to one bit of precision using an ancillary transmon [90].
Honeycomb Floquet code	Plaquette stabilizer measurement realized on the IBM Falcon superconducting-qubit device [66]
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) [91] (see Ref. [92], Sec. 5.6 and Ref. [93], Ch. 4).
Irregular LDPC code	Satellite communication after concatenating with a modulation scheme [94].
Irregular repeat-accumulate (IRA) code	LDPC codes are used for digital satellite video broadcasting per the DVB-S2 standard [95,96] utilize IRA code features and are subject to ongoing litigation; see Ref. [97].Apple and Broadcom Wi-Fi devices utilize IRA encoding and decoding code features and are subject to ongoing litigation; see Ref. [97].
Justesen code	Generating small-bias sample spaces, i.e., probability distributions that parity functions cannot typically distinguish from the uniform distribution [98].
Kerdock code	Digital fingerprinting: $t$-collision secure schemes can be designed to detect a pirated copy once $t$ users have colluded [99].
Kitaev chain code	Photonic systems: braiding of defects has been simulated in a device that has a different notion of locality than a bona-fide fermionic system [100].Superconducting circuits: initialization [101], braiding [102] and detection [102,103] of defects has been simulated in devices that have a different notion of locality than a bona-fide fermionic system.
Kitaev surface code	One cycle of syndrome readout on 19-qubit planar and 24-qubit toric codes realized in two-dimensional Rydberg atomic arrays [104]. Signatures of corresponding topological phase of matter detected in superconducting circuits [105] and two-dimensional Rydberg atomic arrays [106]. Ground state of the toric code has been implemented with and without twists, and the non-Abelian braiding behavior of the twists, which realize Ising anyons, has been demonstrated [107].
Lloyd-Slotine nine-mode code	Optical network by the Furusawa group [108].
Locally recoverable code (LRC)	An $(18,14,7)$ LRC code has beed used in the Windows Azure cloud storage system [109]; see also [110; 31.3.1.2].
Low-rank parity-check (LRPC) code	Cryptosystem [111] that is a rank-metric analogue of NTRU [112] and MDPC [113] cryptosystems.Post-quantum cryptography [114].
Matching code	Braiding of defects has been demonstrated for a five-qubit version of code [115].
Maximum distance separable (MDS) code	The McEliece Public Key Cryptosystem [116] uses algebraic coding theory to secure communications against eavesdropping attack, in which private keys are generator matrices of linear codes, i.e., $G$. Public Keys shared to outside world are scrambled and permutated versions of $G$, i.e., $G^\prime=SGP$. Data to be encrypted, $u$, is multiplied by public key and added intentional errors $e$, i.e., $x=uG^\prime+e$. Upon receiving encrypted data, private key owner can apply inverse permutation $P^{-1}$ to $x$, decode the scrambled message given the presence of $e$ errors, and finally unscramble to obtain $u$. Security parameters of the system are $n$ and $e$, hence MDS codes, such as GRS codes may provide the same security level for shorter key sizes, and the private key owner can decode arguably fast enough using known decoding algorithms.Automatic repeat request (ARQ) data transmission protocols ([93], Ch. 7).
Maximum-rank distance (MRD) code	Useful for error and erasure correction in network coding [117,118].
Monitored random-circuit code	Measurement induced quantum phases have been realized in a trapped-ion processor [119].
NTRU-GKP code	Public-key quantum communication protocol [120].
Niset-Andersen-Cerf code	Realized in Ref. [121] in an optical system with 3 beam-splitters. The fidelity peaked around $0.6$ for deterministic approach, and around $0.77$ for the probabilistic approach (with a 25% chance of error).
On-off keyed (OOK) c-q code	Proof-of-principle experiments using Dolinar [122] and TES receivers [123].
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 ([124], Ch. 27).Marking the state of a finite automaton [125].Used in machine-learning based classification tasks because one-hot encodings, as opposed to integer encodings, do not presume an order [126].
PPM c-q code	Conditional pulse nulling (CPN) receiver [127].
PSK c-q code	Unambiguous state discrimination using displacement-based receiver for 4-PSK [128].Multi-stage quantum receivers [129–132].Bayesian inference [133].Time resolving quantum receiver opertaing in the telecom C band [134].Displacements and photon detection [135].Adaptive decoder using linear-optical elements and photon detection [12].
Pair-cat code	Microwave cavities coupled to superconducting circuits by the Wang group [136].
Phase-shift keying (PSK) code	Telephone-line modems: 1967 Milgo 4400/48 and international standard V.27 used 8-PSK [137].
Polar code	Code control channels for the 5G NR (New Radio) interfaces [138].
Pulse-position modulation (PPM) code	Greek hydraulic semaphore system [139,140].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 [141].Telephone-line modems: 1962 Bell 201 and international standard V.24 [142].
Quadrature-amplitude modulation (QAM) code	Optical communication (e.g., Ref. [143]).Telephone-line modems: 1971 Codex 9600C and international standard V.29 used 16-QAM [144].
Quantum Tanner code	Used to obtain explicit lower bounds in the sum-of-squares game [145].States that, on average, achieve small violations of check operators for quantum Tanner codes require a circuit of non-constant depth to make. They are used in the proof [146] of the No low-energy trivial states (NLTS) conjecture [147].
Quantum parity code (QPC)	The $[[m^2,1,m]]$ codes for $m\leq 7$ have been realized in trapped-ion quantum devices [75].
Quantum repetition code	NMR: 3-qubit phase-flip code [148,149], with up to two rounds of error correction in liquid-state NMR [150].Superconducting circuits: 3-qubit phase-flip and bit-flip code by Schoelkopf group [151]; 3-qubit bit-flip code [152]; 3-qubit phase-flip code up to 3 cycles of error correction [153]; IBM 15-qubit device [154]; IBM Rochester device using 43-qubit code [155]; Google system performing up to 8 error-correction cycles on 5 and 9 qubits [156]; Google Quantum AI Sycamore utilizing up to 11 physical qubits and running 50 correction rounds [157]; Google Quantum AI Sycamore utilizing up to 25 qubits for comparison of logical error scaling with a quantum code [158] (see also [159]).Continuous error correction protocols have been implemented on a 3-qubit superconducting qubit device [160].Semiconductor spin-qubit devices: 3-qubit devices at RIKEN [161] and Delft [162].Nitrogen-vacancy centers in diamond: 3-qubit phase-flip code [163,164] (see also Ref. [165]).Trapped-ion device: 3-qubit phase-flip algorithm implemented in 3 cycles on high fidelity gate operations [166]. Both phase- and bit-flip codes for 31 qubits and their stabilizer measurements have been realized by Quantinuum [167].
Quasi-cyclic LDPC (QC-LDPC) code	5G NR cellular communication for the traffic channel [168,169].Wireless communication: WiMAX (IEEE 802.16e) [170–172], WiFi 4 (IEEE 802.11n) [173], and WPAN (IEEE 802.15.3c) [174].
Qubit CSS code	Fully homomorphic encryption [175].Entanglement purification protocols related to quantum key distribution (QKD) [176].Cryptographic applications stemming from the monogamy of entanglement of CSS code and error words [177].
Qubit c-q code	Quantum enhancement was demonstrated using a polarization-based non-error-correcting c-q encodings [178].
Random code	Distributed storage systems [179].Classical and quantum cryptography based on the learning-with-errors problem, which is related to decoding a random linear code [180].
Rank-metric code	Identity-Based Encryption [181].Digital watermarking [182].Network coding and streaming media broadcasting [183].
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 [184].
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 [185,186]. For example, the $(32, 6, 16)$ RM code was used for the Mariner 9 spacecraft.
Reed-Solomon (RS) code	RS Product Code (RSPC) was used in DVDs (see Ref. [93], Ch. 4).DSL technologies and their variants against impluse noise [187].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 [188].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 temetry channel coding standard issued by the Consultative Committee for Space Data Systems (see Ref. [93], Ch. 3).The ubiquity of RS codes has yielded off-the-shelf VLSI intergrated-circuit decoding hardware [189] (see also Ref. [93], Ch. 5 and 10).Automatic repeat request (ARQ) data transmission protocols (see Ref. [93], Ch. 7).Slow-frequency-hop spread-spectrum transmission (see Ref. [93], Chs. 8-9).Coded sharding designs in blockchains to increase efficiency [190].Private Information Retrieval [191].Used in QR-Codes to retrieve damaged barcodes [192].Wireless communication systems such as 3G, DVB, and WiMAX [193].Correcting pooled testing results for SARS-CoV-2 [194].
Regular binary Tanner code	First hardware implementation was done using a semi-systolic decoding architecture [195].
Repetition code	Repetition codes, in conjunction with other codes, were used in magnetic disks [196].Communication protocol such as FlexRay [197] is using repetition code'
Residue AG code	Improvements over the McEliece public-key cryptosystem to linear AG codes on curves of arbitrary genus [198]. Only the subfield subcode proposal remains resilient to attacks [199; Sec. 15.7.5.3].Algebraic secret-sharing schemes [200].
Single parity-check (SPC) code	Can be realized on almost every communication device. SPCs are some of the earliest error-correcting codes ([124], 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) [201–203].
Square-lattice GKP code	Motional degree of freedom of a trapped ion: square-lattice GKP encoding realized with the help of post-selection by Home group [204,205], followed by realization of reduced form of GKP error correction, where displacement error syndromes are measured to one bit of precision using an ion electronic state [206]. State preparation also realized by Tan group [25].Microwave cavity coupled to superconducting circuits: reduced form of square-lattice GKP error correction, where displacement error syndromes are measured to one bit of precision using an ancillary transmon [90]. Subsequent paper by Devoret group [23] (see also [207]) uses reinforcement learning for error-correction cycle design and is the first to go beyond break-even error-correction, with the lifetime of a logical qubit exceeding the cavity lifetime by about a factor of two (see also [22]). See Ref. [24] for another experiment.GKP states and homodyne measurements have been realized in propagating telecom light by the Furusawa group [208].Single-qubit $Z$-gate has been demonstrated in the single-photon subspace of an infinite-mode space [209], in which time and frequency become bosonic conjugate variables of a single effective bosonic mode.In signal processing, GKP state position-state wavefunctions are related to Dirac combs [210].
Surface-17 code	Implemented at ETH Zurich by the Wallraff group [211] and on the Zuchongzhi 2.1 superconducting quantum processor [212]. Both experimental error rates are above the pseudothreshold for this code relative to a single qubit; see Physics viewpoint for a summary [213]. Magic state have been created on the latter processor [214].
Tensor-product code	Construction can be used in magnetic recording by taking the tensor product of a Reed-Solomon code and a parity-check code [215].
Ternary Golay code	Code used in football pools with at least one good bet [46,216]. In fact, the code was originally constructed by Juhani Virtakallio and published in the Finnish football pool magazine Veikkaaja [46,217].Proofs of the quantum mechanical Kochen-Specker theorem [78].
Two-component cat code	Lindbladian-based [218,219] and Hamiltonian-based 'Kerr-cat' [220] encodings have been achieved in superconducting circuit devices by the Devoret group; Ref. [219] also demonstrated a displacement-based gate. The Lindbladian-based scheme has further achieved a suppression of bit-flip errors that is exponential in the average photon number up to a bit-flip time of 1ms [221] and later 10 seconds [222], with the latter scheme repurposing a holonomic gate [223] as a measurement.T4C code realized in a superconducting circuit device by the Wang group [224].
Unary code	Neural networks [225].Birdsong production [226].
Very small logical qubit (VSLQ) code	Star-code autonomous correction scheme realized using superconducting circuits [227].
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 [217,228].
XZZX surface code	Superconducting circuits: Distance-five 25-qubit code implemented on a superconducting quantum processor by Google Quantum AI [158]. This code outperformed the average of several instances of the smaller distance-three 9-qubit $XZZX$ variant of the surface-17 code realized on the same device, both in terms of logical error probability over 25 cycles and in terms of logical error per cycle. This increase in error-correcting capabilities while using more physical qubits supports the notion of an error threshold. Braiding of defects has been demonstrated for the distance-five code [229]. Leakage errors have been handled in a separate work in a distance-three code [159].
Zero-pi qubit code	A related superconducting circuit has been realized by the Houck group [230].
Zetterberg code	Code used to provide better protection of data transmission with its double error correcting capacity [231].
$A_2$ hexagonal lattice code	Wireless communication [232,233].
$[[2m,2m-2,2]]$ error-detecting code	Trapped-ion devices: 12-qubit device by Quantinuum [234]. Subsequent experiment performed Bayesian Quantum Phase Estimation on the $m=3$ code [235].
$[[4,2,2]]$ CSS code	$[[4,1,2]]$ subcode implemented using four-qubit graph state of photons [236].Trapped-ion device by IonQ [237].Logical state preparation and flag-qubit error correction realized in superconducting-circuit devices by IBM [86,87,238,239].The subcode $\{\|\overline{00}\rangle,\|\overline{10}\rangle\}$ [240] and $\{\|\overline{00}\rangle,\|\overline{01}\rangle\}$ [157], treated as a planar surface code, has been realized in superconducting-circuit devices.Logical gates between two copies of the subcode $\{\|\overline{10}\rangle,\|\overline{11}\rangle\}$, interpreted as lattice surgery between planar surface codes, realized in superconducting circuits [241].Logical gates for the $\{\|\overline{00}\rangle,\|\overline{11}\rangle\}$ subcode, treated as a planar code, realized in superconducting circuits [242].The CZ magic state has been realized on an IBM heavy-hex superconducting circuit device [243].
$[[7,1,3]]$ Steane code	Trapped-ion qubits: seven-qubit device in Blatt group [244], ten-qubit QCCD device by Quantinuum [245] realizing repeated rounds of error correction, real-time look-up-table decoding, and non-fault-tolerant magic-state distillations (see APS Physics Synopsis [246]). Fault-tolerant universal two-qubit gate set by Monz group [247]. Logical CNOT gate between two logical qubits, including rounds of correction and fault-tolerant primitives such as flag qubits and pieceable fault tolerance, on a 20-qubit device by Quantinuum [63]; logical fidelity interval of the combined preparation-CNOT-measurement procedure was higher than that of the unencoded physical qubits.Rydberg atom arrays: Lukin group [104]; transversal CNOT gate performed on distance $3$, $5$, and $7$ codes, logical ten-qubit GHZ state initialized with break-even fidelity, fault-tolerant logical two-qubit GHZ state initialized [248].
$[[8,3,2]]$ CSS code	Trapped ions: one-qubit addition algorithm implemented fault-tolerantly on the Quantinuum H1-1 device [249].Superconducting circuits: fault-tolerant $CZZ$ gate performed on IBM and IonQ devices [250].Rydberg atom arrays: Lukin group [251]. 48 logical qubits, 228 logical two-qubit gates, 48 logical CCZ gates, and error detection peformed in 16 blocks. Circuit outcomes were sampled and cross-entropy (XEB) was calculated to verify quantumness. Logical entanglement entropy was measured [248].
$[[9,1,3]]$ Shor code	Trapped-ion qubits: state preparation with 98.8(1)% and 98.5(1)% fidelity for state $\|\overline{0}\rangle$ and $\|\overline{1}\rangle$, respectively, by N. Linke group [75]. Variants of the code to handle coherent noise studied and realized by K. Brown and C. Monroe groups [252].Optical systems: quantum teleportation of information implemented by J.-W. Pan group on maximally entangled pair of one physical and one logical qubit with fidelity rate of up to 78.6% [253]. All-photonic quantum repeater architecture tested on the same code [254].

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