In coding theory, an error correcting code is a means to encode data in a way that is robust against errors (noise).
Very broadly, for a finite set playing the role of a space of states that is to be saved/communicated/analyzed, an error correcting code for is an injection into a larger set. The idea is that noise/errors move the image of within , but if is large enough and the embedding chosen well enough, then a sufficiently small number of errors stays within a small neighbourhood of in that allows to retract back to .
The simplest example is the repetition code, where the inclusion is the diagonal on the -fold Cartesian product
This code “protects against errors” in an evident sense.
Much attention in coding theory is instead on the special class of linear codes, where and carry the structure of vector spaces (necessarily over a finite field if they are finite sets of relevance in practice) and where the inclusion is a linear map.
See also the references at coding theory and linear code.
An observation on classical codes preconceiving aspects of holographic tensor network quantum error correcting codes:
Construction of chiral 2d SCFTs from error-correcting codes:
On their elliptic genera
Last revised on August 24, 2024 at 12:26:10. See the history of this page for a list of all contributions to it.