Generator matrix

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In coding theory, a generator matrix is a matrix whose rows form a basis for a linear code. The codewords are all of the linear combinations of the rows of this matrix, that is, the linear code is the row space of its generator matrix.

Terminology[edit]

If G is a matrix, it generates the codewords of a linear code C by,

w = s G,

where w is a codeword of the linear code C, and s is any input vector. Both w and s are assumed to be row vectors.[1] A generator matrix for a linear -code has format , where n is the length of a codeword, k is the number of information bits (the dimension of C as a vector subspace), d is the minimum distance of the code, and q is size of the finite field, that is, the number of symbols in the alphabet (thus, q = 2 indicates a binary code, etc.). The number of redundant bits is denoted by r = n - k.

The standard form for a generator matrix is,[2]

,

where is the k×k identity matrix and P is a k×r matrix. When the generator matrix is in standard form, the code C is systematic in its first k coordinate positions.[3]

A generator matrix can be used to construct the parity check matrix for a code (and vice versa). If the generator matrix G is in standard form, , then the parity check matrix for C is[4]

,

where is the transpose of the matrix . This is a consequence of the fact that a parity check matrix of is a generator matrix of the dual code .

Equivalent Codes[edit]

Codes C1 and C2 are equivalent (denoted C1 ~ C2) if one code can be obtained from the other via the following two transformations:[5]

  1. arbitrarily permute the components, and
  2. independently scale by a non-zero element any components.

Equivalent codes have the same minimum distance.

The generator matrices of equivalent codes can be obtained from one another via the following elementary operations:[6]

  1. permute rows
  2. scale rows by a nonzero scalar
  3. add rows to other rows
  4. permute columns, and
  5. scale columns by a nonzero scalar.

Thus, we can perform Gaussian Elimination on G. Indeed, this allows us to assume that the generator matrix is in the standard form. More precisely, for any matrix G we can find a invertible matrix U such that , where G and generate equivalent codes.

See also[edit]

Notes[edit]

  1. ^ MacKay, David, J.C. (2003). Information Theory, Inference, and Learning Algorithms (PDF). Cambridge University Press. p. 9. ISBN 9780521642989. Because the Hamming code is a linear code, it can be written compactly in terms of matrices as follows. The transmitted codeword is obtained from the source sequence by a linear operation,

    where is the generator matrix of the code... I have assumed that and are column vectors. If instead they are row vectors, then this equation is replaced by

    ... I find it easier to relate to the right-multiplication (...) than the left-multiplication (...). Many coding theory texts use the left-multiplying conventions (...), however. ...The rows of the generator matrix can be viewed as defining the basis vectors.
     
  2. ^ Ling & Xing 2004, p. 52
  3. ^ Roman 1992, p. 198
  4. ^ Roman 1992, p. 200
  5. ^ Pless 1998, p. 8
  6. ^ Welsh 1988, pp. 54-55

References[edit]

  • Ling, San; Xing, Chaoping (2004), Coding Theory / A First Course, Cambridge University Press, ISBN 0-521-52923-9 
  • Pless, Vera (1998), Introduction to the Theory of Error-Correcting Codes (3rd ed.), Wiley Interscience, ISBN 0-471-19047-0 
  • Roman, Steven (1992), Coding and Information Theory, GTM, 134, Springer-Verlag, ISBN 0-387-97812-7 
  • Welsh, Dominic (1988), Codes and Cryptography, Oxford University Press, ISBN 0-19-853287-3 

Further reading[edit]

External links[edit]