# Minimum distance of a code given its basis

I know that to determine the minimum distance of a code given its basis we can follow this procedure:

Let $$\{ c_1, \ldots, c_k \}$$ be a basis for a code of length $$n$$ and dimension $$k$$. Then the generator matrix is $$B$$ whose rows are $$c_i$$. I know that row ops on $$B$$ do not change the code so we can assume that $$B = (P | I_k)$$. Now the parity check matrix is given by $$A = (P^\top | I_{n-k})$$ and the minimum distance of the code is the minimum number of columns of $$A$$ that are linearly dependent.

The problem is that this approach can be very tedious for large codes as we have to reduce $$B$$ to echelon form, so my question are:

1) Is there in general an easier approach? Eg. looking at the weight of the codewords in the basis.

2) If $$C$$ is cyclic should I also follow the same approach?

• 1) There is and its implemented in Magma. 2) For specific codes like BCH codes there are better methods due to the structure of the codes. May 5, 2020 at 15:58
• What is Magma??
– john
May 5, 2020 at 16:05
• magma.maths.usyd.edu.au/magma May 5, 2020 at 16:07
• Computer algebra system. May 5, 2020 at 16:07