Several classes of BCH and negacyclic BCH codes of length $n = \frac{q^{m}-1}{2}$
Mengchen Lian, Shixin Zhu · Advances in Mathematics of Communications · 2024
BCH codes are an important class of linear codes and find extensive utilization in communication and disk storage systems. This paper mainly analyzes the negacyclic BCH code and BCH code of length $ \frac{q^m-1}{2} $. For negacyclic BCH code, we give the dimensions of $ \mathcal{C}_{(n, -1, \left\lceil \frac{\delta+1}{2}\right\rceil, 0)} $ for $ \delta = a\frac{q^m-1}{q-1}, aq^{m-1}-1 $($ 1\leq a <\frac{q-1}{2} $) and $ \delta = a\frac{q^m-1}{q-1}+b\frac{q^m-1}{q^2-1}, aq^{m-1}+(a+b)q^{m-2}-1 $ $ (2\mid m, 1\leq a+b \leq q-1 $, $ \left\lceil \frac{q-a-2}{2}\right\rceil\geq 1) $. Furthermore, the dimensions of negacyclic BCH codes $ \mathcal{C}_{(n, -1, \delta, 0)} $ with few nonzeros and $ \mathcal{C}_{(n, -1, \delta, b)} $ with $ b eq 0 $ are settled. For BCH code, we give the weight distribution of extended code $ \overline{ \mathcal{C}}_{(n, 1, \delta, 1)} $ and the parameters of dual code $ \mathcal{C}^{\perp}_{(n, 1, \delta, 1)} $, where $ \delta_2\leq \delta \leq \delta_1 $.