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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== 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 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, : , 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. 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 : , 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 . 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「generator matrix」の詳細全文を読む スポンサード リンク
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