On the parameters of some LCD BCH codes over $$\mathbb {F}_q$$ with length $$(q^m+1)/\lambda $$

Abstract

As a particular subclass of cyclic codes, BCH codes have wide applications in storage devices, communication systems, consumer electronics and other fields. However, parameters of BCH codes are unknown in general. In this paper, we investigate parameters of BCH codes of length \(\frac{q^m+1}{\lambda }\) where \(\lambda \mid q+1\) .Some new techniques are employed to study the coset leaders. For any odd prime power q and \(m=4,8\) , or \(m\ge 12\) and \(m\equiv 4~ (\textrm{mod}~ 8)\) , the second, the third and the fourth largest coset leaders modulo \(q^m+1\) are determined, and the dimensions of some BCH codes of length \(q^m+1\) with large designed distances are given. For \(1<\lambda <q+1\) , the first few largest coset leaders and the coset leaders modulo \(\frac{q^m+1}{\lambda }\) in the range 1 to \( \frac{ q^{\lfloor (m+1)/2\rfloor }}{\lambda }\) are studied, and the dimensions of some BCH codes of length \(\frac{q^m+1}{\lambda }\) are given as well. The BCH codes presented in this paper are LCD codes and have a sharper lower bound on the minimum distance than the well-known BCH bound.