Generator Polynomial and Minimum Distance

Question

Given a generator polynomial, how do I calculate minimum distance for the code. I am working in GF(2).

A particular case of interest is the cyclic code of length $9$ generated by $$ g(x)=x^7+x^6+x^4+x^3+x+1. $$

Answer 1

For general cyclic codes, no efficient way is known to compute the minimum distance. For example, the exact minimum distances of the binary quadratic residue code of lengths > 200 (certain cyclic codes) are not known.

You can apply various bounds (in particular, the BCH bound) to get a rough idea about the value, which may be enough to pin down the minimum distance for small cases. But in general, no better approach is known than enumerating an exponential number of codewords. I've sketched the best known method for general linear codes in this answer.

Answer 2

Any cyclic code is a linear code. So you may find the minimum distance by the following algorithm

1. Find all code polynomials for this code.

2. Calculate a number of terms for each code polynomial.

3. The minimal number of terms be exactly the minimum distance.