Several classes of BCH and negacyclic BCH codes of length $n = \frac{q^{m}-1}{2}$
Year: 2024
Type: article
Abstract: 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\neq 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 $.
DOI: 10.3934/amc.2024049
Authors: Mengchen Lian, Shixin Zhu
Institution: Hefei University of Technology
Language: English
Subfield: Artificial Intelligence
Field: Computer Science
Domain: Physical Sciences
Open Access status: diamond
OpenAlex ID: W4404770294