Here is a list of all classical and quantum codes that have been realized in experimental or real-world devices.
Name Realization(s)
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 [1].
Binary Golay code Used in the Voyager 1 and 2 spacecraft [2].
Binary repetition code Repetition codesm in conjunction with other codes, were used in magnetic disks [3].
Binomial code Realized in microwave cavities coupled to superconducting circuits [4].
Cat code Two-legged ($$S=1$$) cat code has been realized in a superconducting circuit device by the Devoret group [5]. Exponential suppression of bit-flip errors achieved [6] up to a bit-flip time of 1 ms. A bit-flip time of up to 1 sec has been achieved while away from the logical-qubit regime [7].Four-legged ($$S=2$$) cat code has been realized in a superconducting circuit device [8]. This paper is the first to reach break-even error-correction, where the lifetime of a logical qubit is on par with the lifetime of the noisiest constituent of the system.
Convolutional code A type of convolutional code used in Real-time Application networks [9].Mobile and radio communications (3G networks) use convolutional codes concatenated with Reed-Solomon codes to obtain suitable performance [10].A convolutional code with rate 1/2 was used for deep-space and satellite communication [11]
Covering code Data compression both with or without compression [12].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 [13][14].
Fountain code Designed for servers sending data to many recipients, such as during broadcasting or file distribution [15].
Generalized Reed-Solomon (GRS) code Various cloud storage systems [16].Public-key cryptosystems [17]. Initial construction of McEliece Public Key Cryptosystem was based on Goppa codes which are subfield subcode of GRS codes. Public Key Cryptosystem designs based on GRS codes first proposed in Ref. [18], which replaced the generator matrix with the parity check matrix, were proven to be insecure [19] since the public key exposes algebraic structure of code. More recent works focus on methods to mask the algebraic structure using subcodes of GRS codes [17]. Lately a key-recovery attack was developed in [20] for a variant of masking method proposed by [21].
Goppa code The binary version $$(q=2)$$ is commonly used in post-quantum cryptosystems such as the McElise cryptosystem [22].
Gottesman-Kitaev-Preskill (GKP) code GKP encoding realized in the motional degree of freedom of a trapped ion [23], followed by realization of dissipative stabilization scheme [24].A reduced form of GKP error correction, consisting of measuring only the direction of a displacement error with an ancillary transmon, realized in a microwave cavity coupled to superconducting circuits [25].
Hamming code Commonly used when error rates are very low, for example, computer RAM or integrated circuits [26].
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 $$d=2$$ [27][28] and $$d=3$$ [29].
Homological bosonic code No experimental realization. However, Ref. [30] describes a potential experimental optical procedure for the simplest non-trival code with 5 modes.
Justesen code Generating small-bias sample spaces, i.e., probability distributions that parity functions cannot typically distinguish from the uniform distribution [31].
Kitaev surface code Distance-two surface codes have been implemented by Andersen et al. [32], Erhard et al. [33], Marques et al. [34], Google Quantum AI [35], and both planar and toric versions by Bluvstein et al. [36]. Distance-three surface code implemented at ETH Zurich [37] and on the Zuchongzhi 2.1 superconducting quantum processor [38]. Signatures of corresponding topological phase of matter detected in superconducting circuits [39] and two-dimensional arrays of Rydberg atoms [40].
Locally recoverable code (LRC) An $$(18,14,7)$$ LRC code has beed used in the Windows Azure cloud storage system [41]; see also Sec. 31.3.1.2 in Ref. [42].
Low-density parity-check (LDPC) code 5G NR cellular communication for the traffic channel [43].WiMAX (IEEE 802.16e) [44].Satelite transmission of digital television [45].
Maximum distance separable (MDS) code The McEliece Public Key Cryptosystem [46] 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.
Maximum-rank distance (MRD) code Useful for error and erasure correction in network coding [47][48].
Monitored random-circuit code Measurement induced quantum phases have been realized in a trapped-ion processor [49].
Niset-Andersen-Cerf code Realized in Ref. [50] 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).
Polar code Code control channels for the 5G NR (New Radio) interfaces [51].
Quantum Tanner code Used to obtain explicit lower bounds in the sum-of-squares game [52].
Quantum parity code (QPC) The $$[[m^2,1,m]]$$ codes for $$m\leq 7$$ have been realized in trapped-ion quantum devices [53].Non-determinisitic linear-optical encoding [54] whose success probability $$P_{E}$$ is determined by the efficiency $$\eta$$ of the photonic encoding circuit. A threshold $$\eta > 0.82$$ exists for the efficiency, above which $$P_{E}\to 1$$ as $$m_1\to\infty$$ given particular $$m_2$$.Studied in the context of error-corrected quantum repeaters [55].
Quantum repetition code NMR: 3-qubit device [56].Superconducting circuits: IBM 15-qubit device [57], Google Quantum AI Sycamore utilizing from 5 to 21 qubits [35].Semiconductor spin-qubit devices: 3-qubit devices at RIKEN [58] and Delft [59].Nitrogen-vacancy centers in diamond: 3-qubit device [60].Continuous error correction protocols have been implemented on a 3-qubit superconducting circuit device [61].Liquid-state NMR: 3-qubit phase-flip code [56].See Table S6 in Ref. [35] for a history of earlier implementations.Repetition codes can be used to benchmark device performance [62].
Random code Distributed storage systems [63].
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 [64].
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-Solomon (RS) code Cross-interleaved RS code (CIRC) is adopted in Compact Discs (CDs) and RS Product Code (RSPC) in DVDs; see Ch. 4 of Ref. [65].In DSL technologies and its variants against impluse noise [66].RS codes as outer codes concatenated with convolutional codes are used indirectly in solar exploration programs; see Ch. 3 of Ref. [65].Coded sharding designs in blockchains to increase efficiency [67].Private Information Retrieval [68].Used in QR-Codes to retrieve damaged barcodes [69].Wireless communication systems such as 3G, DVB, and WiMAX [70].Correcting pooled testing results for SARS-CoV-2 [71].
Shor $$[[9,1,3]]$$ 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 [53]. Variants of the code to handle coherent noise studied and realized by K. Brown and C. Monroe groups [72].All-photonic quantum repeater architecture [73].
Single parity-check code Can be realized on almost every communication device.
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) [74][75].
Steane $$[[7,1,3]]$$ code Trapped-ion qubits: seven-qubit device in Blatt group [76], ten-qubit QCCD device by Quantinuum [77], fault-tolerant universal two-qubit gate realized by Monz group [78].Rydberg atom arrays: Lukin group [36].
Zetterberg code Code used to provide better protection of data transmission with its double error correcting capacity [79].
$$[[4,2,2]]$$ CSS code Trapped-ion device by IonQ [80].Logical state preparation and flag-qubit error correction realized in superconducting-circuit devices by IBM [27][28].
$$[[5,1,3]]$$ perfect code First realized in NMR [81].Demonstration with superconducting qubits [82].

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