Dimension of nonbinary antiprimitive BCH codes

Bose-Chaudhuri-Hocquenghem (BCH) codes have been widely employed in satellite communications, compact disc players, DVDs, disk drives, solid-state drives, two-dimensional bar codes and in cryptography more recently. However, there is only a little known about primitive BCH codes, let alone nonprimitive ones. In this paper, dimension of a special class of nonprimitive BCH codes of length $n=q^{m}+1$ ( which are also called antiprimitive BCH codes) are studied. Some new approaches, such as iterative algorithm, partition and scaling, are adopted to determine the first several largest coset leaders modulo $n=q^{2t+1}+1$ along with coset leaders of $C_{x}$ modulo $n=q^{m}+1$ for $q^{\lceil \frac{m}{2}\rceil}<x<2(q^{\lceil \frac{m}{2} \rceil}+q)$. After deriving the cardinalities of these cyclotomic cosets, we shall calculate precisely dimension of some antiprimitive BCH codes.