On explicit minimum weight bases for extended cyclic codes related to Gold functions

Minimum weight bases of some extended cyclic codes can be chosen from the affine orbits of certain explicitly represented minimum weight codewords. We find such bases for the following three classes of codes: the extended primitive 2-error correcting BCH code of length \(n=2^m,\) where \(m\ge 4\) (for \(m\ge 20\) the result was proven in Grigorescu and Kaufman IEEE Trans Inf Theory 58(I. 2):78–81, 2011), the extended cyclic code \(\bar{C}_{1,5}\) of length \(n=2^m,\) odd m,  \(m\ge 5,\) and the extended cyclic codes \(\bar{C}_{1,2^i+1}\) of lengths \(n=2^m,\) \((i,\,m)=1\) and \(3\le i\le \frac{m-5}{4}-o(m).\)