Error-correcting code (ECC) 


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


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 [24], moderate [57] and large [811] deviation analysis. Sometimes the difference from the asymptotic rate at finite block length can be characterized by the channel dispersion [4,12].


See Ref. [13] for a list of open problems in coding theory.See Refs. [14,15] for reviews of coding theory.


  • Operator-algebra error-correcting 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 what codewords are functions on the alphabet. For example, a bit of information can be embedded into a two-dimensional vector space by associating the two bit values with two basis vectors for the space.



  • Quantum error-correcting code (QECC) — Quantum information cannot be copied using a linear process [17], so one cannot send several copies of a quantum state through a channel as can be done for classical information. Error-correction conditions can similarly be formulated for classical codes [18; Sec. 3], although they are not as widely as used as those for quantum codes.


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“Error-correcting code (ECC)”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.
@incollection{eczoo_ecc, title={Error-correcting code (ECC)}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={} }
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“Error-correcting code (ECC)”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.