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. Linear codes allow for more efficient encoding and decoding algorithms than other codes (cf. syndrome decoding).
Contents
- Definition and parameters
- Generator and check matrices
- Example Hamming codes
- Example Hadamard codes
- Nearest neighbor algorithm
- Popular notation
- Singleton bound
- Examples
- Generalization
- References
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. 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. This code contains 24=16 codewords.
Definition and parameters
A linear code of length n and rank k is a linear subspace C with dimension k of the vector space
The weight of a codeword is the number of its elements that are nonzero and the distance between two codewords is the Hamming distance between them, that is, the number of elements in which they differ. The distance d of a linear code is the minimum weight of its nonzero codewords, or equivalently, the minimum distance between distinct codewords. A linear code of length n, dimension k, and distance d is called an [n,k,d] code.
We want to give
Generator and check matrices
As a linear subspace of
A matrix H representing a linear function
Linearity guarantees that the minimum Hamming distance d between a codeword c0 and any of the other codewords c ≠ c0 is independent of c0. This follows from the property that the difference c − c0 of two codewords in C is also a codeword (i.e., an element of the subspace C), and the property that d(c, c0) = d(c − c0, 0). These properties imply that
In other words, in order to find out the minimum distance between the codewords of a linear code, one would only need to look at the non-zero codewords. The non-zero codeword with the smallest weight has then the minimum distance to the zero codeword, and hence determines the minimum distance of the code.
The distance d of a linear code C also equals the minimum number of linearly dependent columns of the check matrix H.
Proof: Because
Example: Hamming codes
As the first class of linear codes developed for error correction purpose, the Hamming codes has been widely used in digital communication systems. For any positive integer
Example : The linear block code with the following generator matrix and parity check matrix is a
Example: Hadamard codes
Hadamard code is a
Example : The linear block code with the following generator matrix is a
Hadamard code is a special case of Reed-Muller code. If we take the first column (the all-zero column) out from
Nearest neighbor algorithm
The parameter d is closely related to the error correcting ability of the code. The following construction/algorithm illustrates this (called the nearest neighbor decoding algorithm):
Input: A "received vector" v in
Output: A codeword w in C closest to v.
Note: "fail" is not returned unless t > (d − 1)/2. We say that a linear C is t-error correcting if there is at most one codeword in Bt(v), for each v in
Popular notation
Codes in general are often denoted by the letter C, and a code of length n and of rank k (i.e., having k code words in its basis and k rows in its generating matrix) is generally referred to as an (n, k) code. Linear block codes are frequently denoted as [n, k, d] codes, where d refers to the code's minimum Hamming distance between any two code words.
(The [n, k, d] notation should not be confused with the (n, M, d) notation used to denote a non-linear code of length n, size M (i.e., having M code words), and minimum Hamming distance d.)
Singleton bound
Lemma (Singleton bound): Every linear [n,k,d] code C satisfies
A code C whose parameters satisfy k+d=n+1 is called maximum distance separable or MDS. Such codes, when they exist, are in some sense best possible.
If C1 and C2 are two codes of length n and if there is a permutation p in the symmetric group Sn for which (c1,...,cn) in C1 if and only if (cp(1),...,cp(n)) in C2, then we say C1 and C2 are permutation equivalent. In more generality, if there is an
Lemma: Any linear code is permutation equivalent to a code which is in standard form.
Examples
Some examples of linear codes include:
Generalization
Hamming spaces over non-field alphabets have also been considered, especially over finite rings (most notably over Z4) giving rise to modules instead of vector spaces and ring-linear codes (identified with submodules) instead of linear codes. The typical metric used in this case the Lee distance. There exist a Gray isometry between
More recently, some authors have referred to such codes over rings simply as linear codes as well.