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).[citation needed]
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 24=16 codewords.
^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{{cite book}}: CS1 maint: multiple names: authors list (link)
^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.
In coding theory, a linearcode is an error-correcting code for which any linear combination of codewords is also a codeword. Linearcodes are traditionally...
Linear predictive coding (LPC) is a method used mostly in audio signal processing and speech processing for representing the spectral envelope of a digital...
linear network coding is a program in which intermediate nodes transmit data from source nodes to sink nodes by means of linear combinations. Linear network...
described both the linear and bull's eye printing patterns, as well as the mechanical and electronic systems needed to read the code. The patent was issued...
computer science and telecommunication, Hamming codes are a family of linear error-correcting codes. Hamming codes can detect one-bit and two-bit errors, or...
codes, Hadamard codes, Expander codes, Golay codes, Reed–Muller codes and Polar codes. These examples also belong to the class of linearcodes, and hence they...
mathematician Joseph Leonard Walsh. The Hadamard code is an example of a linearcode of length 2 m {\displaystyle 2^{m}} over a binary alphabet. Unfortunately...
contains Linear A Unicode characters. Without proper rendering support, you may see question marks, boxes, or other symbols instead of Linear A. Linear A is...
Golay code is a type of linear error-correcting code used in digital communications. The binary Golay code, along with the ternary Golay code, has a...
PVC. There are further coverage criteria, which are used less often: LinearCode Sequence and Jump (LCSAJ) coverage a.k.a. JJ-Path coverage – has every...
In coding theory, the dual code of a linearcode C ⊂ F q n {\displaystyle C\subset \mathbb {F} _{q}^{n}} is the linearcode defined by C ⊥ = { x ∈ F q...
the code. There are many types of linear block codes, such as Cyclic codes (e.g., Hamming codes) Repetition codes Parity codes Polynomial codes (e.g...
Linearcode sequence and jump (LCSAJ), in the broad sense, is a software analysis method used to identify structural units in code under test. Its primary...
{\displaystyle n-1} cyclic left shifts, a cyclic code may also be defined via cyclic left shifts. Therefore, the linearcode C {\displaystyle {\mathcal {C}}} is cyclic...
Warped linear predictive coding (warped LPC or WLPC) is a variant of linear predictive coding in which the spectral representation of the system is modified...
(1953). Linear block codes that achieve equality in the Singleton bound are called MDS (maximum distance separable) codes. Examples of such codes include...
In coding theory, a generator matrix is a matrix whose rows form a basis for a linearcode. The codewords are all of the linear combinations of the rows...
In coding theory, a polynomial code is a type of linearcode whose set of valid code words consists of those polynomials (usually of some fixed length)...
codeLinear Network Coding, a type of erasure correcting code across networks instead of point-to-point links Long code Low-density parity-check code...
Algebraic geometry codes, often abbreviated AG codes, are a type of linearcode that generalize Reed–Solomon codes. The Russian mathematician V. D. Goppa...
A Sparse graph code is a code which is represented by a sparse graph. Any linearcode can be represented as a graph, where there are two sets of nodes...
In coding theory, the weight enumerator polynomial of a binary linearcode specifies the number of words of each possible Hamming weight. Let C ⊂ F 2...
remote site at a later time. Every non-systematic linearcode can be transformed into a systematic code with essentially the same properties (i.e., minimum...
Global Information
