Dimensions of nonbinary antiprimitive BCH codes and some conjectures

TL;DR

This paper investigates the dimensions of non-primitive antiprimitive BCH codes, introduces new methods to determine coset leaders, and proposes conjectures to facilitate their complete characterization.

Contribution

It provides new approaches to identify coset leaders and derives exact dimensions for antiprimitive LCD BCH codes, advancing understanding of non-primitive BCH codes.

Findings

All coset leaders in a specific range are determined for certain parameters.

The first several largest coset leaders are identified for odd m.

Conjectures about coset leaders are proposed to simplify their determination.

Abstract

Bose-Chaudhuri-Hocquenghem (BCH) codes have been intensively investigated. Even so, there is only a little known about primitive BCH codes, let alone non-primitive ones. In this paper, let $q>2$ be a prime power, the dimension of a family of non-primitive BCH codes of length $n=q^{m}+1$ (also called antiprimitive) is studied. These codes are also linear codes with complementary duals (called LCD codes). Through some approaches such as iterative algorithm, partition and scaling, all coset leaders of $C_{x}$ modulo $n$ with $q^{\lceil \frac{m}{2}\rceil}<x\leq 2q^{\lceil\frac{m}{2} \rceil}+2$ are given for $m\geq 4$. And for odd $m$ the first several largest coset leaders modulo $n$ are determined. Furthermore, a new kind of sequences is introduced to determine the second largest coset leader modulo $n$ with $m$ even and $q$ odd. Also, for even $m$ some conjectures about the first several coset leaders modulo $n$ are proposed, whose complete verification would wipe out the difficult problem to determine the first several coset leaders of antiprimitive BCH codes. After deriving the cardinalities of the coset leaders, we shall calculate exact dimensions of many antiprimitive LCD BCH codes.