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I've gone to Wikipedia and watched a few videos regarding Linear Codes, and from what I see, the Parity Check Matrix is always given, and because of that the Generator Matrix (consisting of Identity Matrix, Parity Matrix) can be made. But what if the Parity Matrix isn't given, and instead, only k and n? I understand the general layout, but I do not see how to figure out the actual contents (or identity, I believe its called) of the Parity Matrix.

  • I should add that the specific scenario I am asking about is in binary, so maybe its only doable there?
rostendalve
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  • is the answer satisfactory? – kodlu Jun 11 '23 at 16:21
  • I am unsure, am I just supposed to pick a Random Matrix A for the Parity matrix? I don't understand. – rostendalve Jun 20 '23 at 15:13
  • There is not enough info in the question to pick a matrix. You need to specify more, like minimum distance of the code. Or pick from a family of codes like BCH, Reed-Muller, Hamming. – kodlu Jun 20 '23 at 15:43

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There is no way to uniquely determine a linear code in general given only $n,k$. This is still true even if it is binary.

Any $k-$ dimensional subspace of the $n$ dimensional vector space $\mathbb{F}_2^n$ forms a code and any basis for that can be taken to be the generator matrix.

The number of $k-$dimensional subspaces of the $n-$dimensional space is known and is given here. If the field is nonprime, with fieldsize $q=p^n$ just replace $p$ with $q.$

Also among the codes with dimension $k,$ there are in general very good ones [high minimum distance] and terrible ones with low minimum distance.

kodlu
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