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.
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 vector. A generator matrix for an (, , )-code has dimensions k×n, 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,
Codes C1 and C2 are equivalent (denoted C1 ~ C2) if one code can be obtained from the other via the following two transformations:
- arbitrarily permute the components, and
- independently scale by a non-zero element any components.
Equivalent codes have the same minimum distance.
- permute rows
- scale rows by a nonzero scalar
- add rows
- permute columns, and
- 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.
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