Cited By
View all- Sun ZLiu XZhu STang Y(2024)Negacyclic BCH codes of length and their dualsDesigns, Codes and Cryptography10.1007/s10623-024-01380-392:7(2085-2101)Online publication date: 1-Jul-2024
The Hermitian Dual Codes of Several Classes of BCH Codes
Pages 4484 - 4497
Abstract
As a special subclass of cyclic codes, BCH codes are usually among the best cyclic codes and have wide applications in communication and storage systems and consumer electronics. Let <inline-formula> <tex-math notation="LaTeX">$\mathcal C$ </tex-math></inline-formula> be a <inline-formula> <tex-math notation="LaTeX">$q^{2}$ </tex-math></inline-formula>-ary BCH code of length <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> with respect 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> over an extension field of <inline-formula> <tex-math notation="LaTeX">$\Bbb F_{q^{2}}$ </tex-math></inline-formula>, and let <inline-formula> <tex-math notation="LaTeX">$\mathcal C^{\perp H}$ </tex-math></inline-formula> denote its Hermitian dual code, where <inline-formula> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula> is a prime power. If both <inline-formula> <tex-math notation="LaTeX">$\mathcal {C}$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$\mathcal C^{\perp H}$ </tex-math></inline-formula> are a BCH code with respect 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>, then <inline-formula> <tex-math notation="LaTeX">$\mathcal C$ </tex-math></inline-formula> is called a <italic>Hermitian dually-BCH code</italic>. The objective of this paper is to derive a necessary and sufficient condition for ensuring that two classes of narrow-sense BCH codes are Hermitian dually-BCH codes. As by-products, lower bounds on the minimum distances of the Hermitian dual codes of these BCH codes are developed, which improve the lower bounds documented in IEEE Trans. Inf. Theory, vol. 68, no. 2, pp. 953-964, 2022, in some cases.
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View all- Sun ZLiu XZhu STang Y(2024)Negacyclic BCH codes of length and their dualsDesigns, Codes and Cryptography10.1007/s10623-024-01380-392:7(2085-2101)Online publication date: 1-Jul-2024
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