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❪∂₂'❫ Linear Codes ○｜Definition｜1st｜20260122191347-00-⌔
Linear code
In coding theory, a linear code is an error-correcting code for which any linear combination of codewords is also a codeword. Linear codes are traditionally partitioned into block codes and convolutional codes, although turbo codes can be seen as a hybrid of these two types.1 Linear codes allow for more efficient encoding and decoding algorithms than other codes (cf. syndrome decoding).
Linear codes are used in forward error correction and are applied in methods for transmitting symbols (e.g., bits) on a communications channel so that, if errors occur in the communication, some errors can be corrected or detected by the recipient of a message block. The codewords in a linear block code are blocks of symbols that are encoded using more symbols than the original value to be sent.2 A linear code of length n transmits blocks containing n symbols. For example, the [7,4,3] Hamming code is a linear binary code which represents 4-bit messages using 7-bit codewords. Two distinct codewords differ in at least three bits. As a consequence, up to two errors per codeword can be detected while a single error can be corrected.3 This code contains 2 = 16 codewords.
Printed 2026-06-28.
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Link to original Footnotes
William E. Ryan and Shu Lin (2009). Channel Codes: Classical and Modern. Cambridge University Press. p. 4. ISBN 978-0-521-84868-8. ↩
MacKay, David, J.C. (2003). Information Theory, Inference, and Learning Algorithms (PDF). Cambridge University Press. p. 9. Bibcode:2003itil.book…M. ISBN 9780521642989. In a linear block code, the extra bits are linear functions of the original bits; these extra bits are called parity-check bits ↩
Thomas M. Cover and Joy A. Thomas (1991). Elements of Information Theory. John Wiley & Sons, Inc. pp. 210–211. ISBN 978-0-471-06259-2. ↩
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