On bases of BCH codes with designed distance 3 and their extensions (Q2229574)

A codeword \(c\) of an affine-invariant code is said to be a single orbit affine generator if the orbit of \(c\) under the action of the affine group of \(\mathbb{F}_{p^m}\) contains a basis of the code. Denote by \(C_{i_1,\ldots, i_\ell}\) the cyclic code \(\{c(x) \mid c(\alpha_j) = 0, j \in \text{cl}(i_1)\cup \cdots \cup \text{cl}(i_\ell)\},\) where \(i_1,\ldots , i_\ell\) are representatives of some cyclotomic classes. In this work, the authors analyze the question of existence of a single orbit affine generator of the minimum possible weight for the extended BCH code over nonbinary prime finite field. It is proved that for any prime \(p\neq 2,3\) neither the code \(C_{1,2}\) nor its extended code \(\overline{C_{1,2}}\) are generated by the codewords of nonzero weight. Also, the nonexistence of bases consisting of minimum weight vectors for extended BCH codes with designed distance 3 over \(\mathbb{F}_p\) for any prime \(p,\) \(p\neq 2, 3\) is shown. It is proved that the rank of a single orbit affine generator of weight 5 of the extended BCH code with designed distance 3 is 3. The obtained constraint on the rank is implicitly used for finding a suitable single orbit affine generator. For \(p = 3\), a single orbit affine generator is of the minimum weight, since the code distance is 5, and for \(p>3,\) it is of the next-to-minimum weight 5.