A class of BCH codes with length $$\frac{q^{2m}-1}{q+1}$$

Abstract

As an important class of cyclic codes, BCH codes are widely employed in satellite communications, DVDs, CD, DAT etc. In this paper, we determine the dimension of BCH codes of length \(\frac{q^{2m}-1}{q+1}\) over the finite fields \({\mathbb {F}}_q\) . We settle a conjecture about the largest q-cyclotomic coset leader modulo n which was proposed by Wu et al. We also get the second largest q-cyclotomic coset leader modulo n if m is odd. Moreover, we investigate the parameters of \({\mathcal {C}}_{(n,q,\delta _i)}\) ( \({\mathcal {C}}_{(n,q,\delta _i)}^\perp\) ) for \(i=1,2\) and \({\mathcal {C}}_{(n,q,\delta )}\) ( \({\mathcal {C}}_{(n,q,\delta )}^\perp\) ) for \(2\le \delta \le q-1\) . What’s more, we obtain many (almost) optimal codes.