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Finding generator matrix for binary linear code given parity check matrix. Ask Question Asked 8 years, ...
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 ...
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.
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.
2. Yes, you can find the generator matrix through finding the kernel of H H, but you can find it through writing H H in its standard form, i.e., H =[A|In×k] H = [A | I n × k]. Then you use A A to build your generator matrix (in its standard form): G =[Ik|At] G = [I k | A t], where At A t is the transpose of the matrix A A. I am working with ...
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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.
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 ...
For example a generator matrix of a binary Simplex code of dimension $4$ is given by $$\begin{pmatrix} 1 ...