The Dual Codes of Several Classes of BCH Codes
Binkai Gong, Cunsheng Ding, Chengju Li
Abstract
As a special subclass of cyclic codes, BCH codes have wide applications in communication and storage systems. A BCH code of length <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> over <inline-formula> <tex-math notation="LaTeX">$\mathbb {F}_{q}$ </tex-math></inline-formula> is always relative to an <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>-th primitive root of unity <inline-formula> <tex-math notation="LaTeX">$\beta $ </tex-math></inline-formula> in an extension field of <inline-formula> <tex-math notation="LaTeX">$\mathbb {F}_{q}$ </tex-math></inline-formula>, and is called a dually-BCH code if its dual is also a BCH code relative to the same <inline-formula> <tex-math notation="LaTeX">$\beta $ </tex-math></inline-formula>. The question as to whether a BCH code is a dually-BCH code is in general very hard to answer. In this paper, an answer to this question for primitive narrow-sense BCH codes and projective narrow-sense ternary BCH codes is given. Sufficient and necessary conditions in terms of the designed distances <inline-formula> <tex-math notation="LaTeX">$\delta $ </tex-math></inline-formula> will be presented to ensure that these BCH codes are dually-BCH codes. In addition, the parameters of the primitive narrow-sense BCH codes and their dual codes are investigated. Some lower bounds on minimum distances of the dual codes of primitive and projective narrow-sense BCH codes are developed. Especially for binary primitive narrow-sense BCH codes, the new bounds on the minimum distances of the dual codes improve the classical Sidel’nikov bound, and are also better than the Carlitz and Uchiyama bound for large designed distances <inline-formula> <tex-math notation="LaTeX">$\delta $ </tex-math></inline-formula>. The question as to what subclasses of cyclic codes are BCH codes is also answered to some extent. As a byproduct, the parameters of some subclasses of cyclic codes are also investigated.