Root code for the Classical Domain

Codes for communication over classical channels

## Description

Code designed for transmission of classical information through classical channels in a robust way.

A code is a subset of a set or alphabet, with each element called a codeword. An error-correcting code consists of \(K\) codewords over an alphabet with \(N\) elements such that it is possible to recover the codewords from errors \(E\) from some error set \(\mathcal{E}\).

## Rate

The Shannon channel capacity (the maximum of the mutual information over input and output distributions) is the highest rate of information transmission through a classical (i.e., non-quantum) channel with arbitrarily small error rate [1]. Corrections to the capacity and tradeoff between decoding error, code rate and code length are determined using small [2–4], moderate [5–7] and large [8–11] deviation analysis. Sometimes the difference from the asymptotic rate at finite block length can be characterized by the channel dispersion [4,12].

## Notes

See Ref. [13] for a list of open problems in coding theory.See Refs. [14,15] for reviews of coding theory.Classical systems such as RAM [16], HPC systems [17–19], and data centers [20] suffer from noise.

## Parent

- Classical-quantum (c-q) code — Any ECC can be embedded into a quantum Hilbert space, and thus passed through a quantum channel, by associating elements of the alphabet with basis vectors in a Hilbert space over the complex numbers. In other words, classical codewords are elements of an alphabet, while quantum codewords are functions on the alphabet. Classical codes can be unified with quantum codes using various algebraic frameworks [21,22].

## Children

- Group-alphabet code
- Block code
- Generalized concatenated code (GCC)
- Parallel concatenated code
- Finite-dimensional error-correcting code (ECC)
- Group-orbit code — Not all codes are group-orbit codes, and more generally one can classify codewords into orbits of the automorphism group [23].
- Random code

## Cousin

- Quantum error-correcting code (QECC) — Quantum information cannot be copied using a linear process [24], so one cannot send several copies of a quantum state through a channel as can be done for classical information. The Knill-Laflamme conditions can similarly be formulated for classical codes [25; Sec. 3], although they are not as widely as used as those for quantum codes.

## References

- [1]
- C. E. Shannon, “A Mathematical Theory of Communication”, Bell System Technical Journal 27, 379 (1948) DOI
- [2]
- V. Strassen, “Asymptotische Absch¨atzungen in Shannons Informationstheorie,” Trans. Third Prague Conference on Information Theory, Prague, 689–723, (1962)
- [3]
- M. Hayashi, “Information Spectrum Approach to Second-Order Coding Rate in Channel Coding”, IEEE Transactions on Information Theory 55, 4947 (2009) arXiv:0801.2242 DOI
- [4]
- Y. Polyanskiy, H. V. Poor, and S. Verdu, “Channel Coding Rate in the Finite Blocklength Regime”, IEEE Transactions on Information Theory 56, 2307 (2010) DOI
- [5]
- Y. Altug and A. B. Wagner, “Moderate Deviations in Channel Coding”, (2012) arXiv:1208.1924
- [6]
- Y. Polyanskiy and S. Verdu, “Channel dispersion and moderate deviations limits for memoryless channels”, 2010 48th Annual Allerton Conference on Communication, Control, and Computing (Allerton) (2010) DOI
- [7]
- C. T. Chubb, V. Y. F. Tan, and M. Tomamichel, “Moderate Deviation Analysis for Classical Communication over Quantum Channels”, Communications in Mathematical Physics 355, 1283 (2017) arXiv:1701.03114 DOI
- [8]
- R. Gallager, Information Theory and Reliable Communication (Springer Vienna, 1972) DOI
- [9]
- I. Csiszár and J. Körner, Information Theory (Cambridge University Press, 2011) DOI
- [10]
- S. Arimoto, “On the converse to the coding theorem for discrete memoryless channels (Corresp.)”, IEEE Transactions on Information Theory 19, 357 (1973) DOI
- [11]
- G. Dueck and J. Korner, “Reliability function of a discrete memoryless channel at rates above capacity (Corresp.)”, IEEE Transactions on Information Theory 25, 82 (1979) DOI
- [12]
- S. H. Hassani, K. Alishahi, and R. L. Urbanke, “Finite-Length Scaling for Polar Codes”, IEEE Transactions on Information Theory 60, 5875 (2014) DOI
- [13]
- S. Dougherty, J.-L. Kim, and P. Solé, “Open Problems in Coding Theory”, Noncommutative Rings and Their Applications 79 (2015) DOI
- [14]
- A. R. Calderbank, “The art of signaling: fifty years of coding theory”, IEEE Transactions on Information Theory 44, 2561 (1998) DOI
- [15]
- D. J. Costello Jr. and G. D. Forney Jr, “Channel Coding: The Road to Channel Capacity”, (2006) arXiv:cs/0611112
- [16]
- B. Schroeder, E. Pinheiro, and W.-D. Weber, “DRAM errors in the wild”, Communications of the ACM 54, 100 (2011) DOI
- [17]
- S. Levy et al., “Lessons Learned from Memory Errors Observed Over the Lifetime of Cielo”, SC18: International Conference for High Performance Computing, Networking, Storage and Analysis (2018) DOI
- [18]
- K. B. Ferreira et al., “Understanding Memory Failures on a Petascale Arm System”, Proceedings of the 31st International Symposium on High-Performance Parallel and Distributed Computing (2022) DOI
- [19]
- C. Di Martino et al., “Lessons Learned from the Analysis of System Failures at Petascale: The Case of Blue Waters”, 2014 44th Annual IEEE/IFIP International Conference on Dependable Systems and Networks (2014) DOI
- [20]
- J. Meza et al., “Revisiting Memory Errors in Large-Scale Production Data Centers: Analysis and Modeling of New Trends from the Field”, 2015 45th Annual IEEE/IFIP International Conference on Dependable Systems and Networks (2015) DOI
- [21]
- M. K. Patra and S. L. Braunstein, “An algebraic framework for information theory: Classical Information”, (2009) arXiv:0910.1536
- [22]
- G. Kuperberg and N. Weaver, “A von Neumann algebra approach to quantum metrics”, (2010) arXiv:1005.0353
- [23]
- J. H. Conway and N. J. A. Sloane, “Orbit and coset analysis of the Golay and related codes”, IEEE Transactions on Information Theory 36, 1038 (1990) DOI
- [24]
- W. K. Wootters and W. H. Zurek, “A single quantum cannot be cloned”, Nature 299, 802 (1982) DOI
- [25]
- B. Yoshida, “Decoding the Entanglement Structure of Monitored Quantum Circuits”, (2021) arXiv:2109.08691

## Page edit log

- Victor V. Albert (2022-11-06) — most recent

## Cite as:

“Error-correcting code (ECC)”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/ecc

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/classical/ecc.yml.