Generatormatrices

Generator matrices, or Generatormatrices, are central objects in the theory of linear codes. Let C be a linear code of length n over a finite field F_q with dimension k. A generator matrix G is a k-by-n matrix whose rows span C. Equivalently, the code consists of all linear combinations c = mG where m ranges over F_q^k.

The row space of G equals the code C. Any two generator matrices that generate the same code are related by left multiplication by an invertible k×k matrix; this corresponds to a change of basis in the message space. Column operations that permute coordinates correspond to permuting the codeword coordinates. The code C is the row space of G, and a parity-check matrix H satisfies G H^T = 0; equivalently, C = { c in F_q^n : H c^T = 0 }.

Systematic form and encoding: Every generator matrix can be transformed into systematic form G = [I_k | P] by elementary row operations and column permutations. In systematic form, encoding is straightforward: c = mG = [m | mP], where m is the message vector in F_q^k. A generator matrix may be converted into a different but equivalent form without changing the underlying code.

Example: Over GF(2), a [5,3] code with G in systematic form is G = [1 0 0 1 0; 0 1 0 1 1; 0 0 1 0 1]. For m = (1,0,1), c = (1,0,1,1,1). This illustrates the encoding process as a linear combination of the rows of G.

Generator matrices are not unique; they are representations of the same code and are used for encoding, code construction, and relation to parity-check matrices and dual codes.