给定码字时，如何找到生成器矩阵

Question

Code = { 000, 011 , 101, 110 }

generator matrix = {011, 101} 

Here the code matrix is given, and I am supposed to find the generator matrix. But I am clueless in determining the generator matrix. Could someone please shed some light on this?

Answer 1

I am not entirely sure what you mean by "generator matrix" in this specific case, but it seems you are looking for a (minimum?) subset B of the code vectors such that each codeword can be represented as a linear combination of the vectors in B.

I suggest you do the following: put your code vectors into a matrix A such that each code vector is a column of A. Then use Gaussian elimination to put A in upper triangular form. The first couple of vectors (the first rank(A) columns to be precise) constitute what I believe you call a generator.

Edit: Let me clarify. If your code lives in GF(2), you must perform Gauss elimination over GF(2) of course.