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As pointed out by Snowball, the problem is inherently hard, see here and also here.. However, it can be done much faster in general than generating all the codewords.
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For example a generator matrix of a binary Simplex code of dimension $4$ is given by $$\begin{pmatrix} 1 ...
2. The generator matrix is only for linear codes, that page is for general codes. For linear binary codes, according to this table, d = 5 d = 5 is the most you can get for n = 20 n = 20, k = 11 k = 11. Share. Cite.
Finding generator matrix for binary linear code given parity check matrix. Ask Question Asked 8 years, ...
1. From your details, your code is binary (you should mention that). You always have that G2 generates some code (its row space). To show that it is a generator matrix for that code, you need to show that it has full row rank. You know that [G1 G2] has full row rank 5. To show that G2 also has full row rank, consider starting with M in standard ...
First, we need to understand what the weight of a codeword is, in terms of S. To find the weight of (1), index the columns of G with two-element subsets of {1, …, m}, and ask what would be the parity of the entry of (1) corresponding to the subset {i, j}. The row vector rs has a 1 at column {i, j} if and only if either i = s or j = s.
3. Given a set C C of codewords, before we can construct a generator matrix, we need to verify that C C is a linear subspace - ie, the sum (and also scalar multiples in the non-binary case) of any two codewords must be a codeword. In the link given, the subsets C C given are all subspaces. The rows of the generator matrix G G can be taken to be ...
2. Yes, you can find the generator matrix through finding the kernel of H, but you can find it through writing H in its standard form, i.e., H = [A | In × k]. Then you use A to build your generator matrix (in its standard form): G = [Ik | At], where At is the transpose of the matrix A. I am working with the same Goppa code and I got this ...
The generator matrix has dimensions k × n k × n, where k k is the dimension of the code and n n is the length of the code. A couple options for finding the parity check matrix: If a code has generator matrix G =(I|A) G = (I | A), it has parity check matrix H =(−AT|I) H = (− A T | I). You can use Gaussian elimination to get G G into the ...