Dimensions of nonbinary antiprimitive BCH codes and some conjectures

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.Comment: Some corrections to version 1, some proofs extended, some critical conjectures propose