Narrow-Sense BCH Codes Over $ {\mathrm {GF}}(q)$ With Length $n=\frac {q^{m}-1}{q-1}$

Cyclic codes are widely employed in communication systems, storage devices, and consumer electronics, as they have efficient encoding and decoding algorithms. BCH codes, as a special subclass of cyclic codes, are in most cases among the best cyclic codes. A subclass of good BCH codes are the narrow-sense BCH codes over <inline-formula> <tex-math notation="LaTeX">$ {\mathrm {GF}}(q)$ </tex-math></inline-formula> with length <inline-formula> <tex-math notation="LaTeX">$n=(q^{m}-1)/(q-1)$ </tex-math></inline-formula>. Little is known about this class of BCH codes when <inline-formula> <tex-math notation="LaTeX">$q>2$ </tex-math></inline-formula>. The objective of this paper is to study some of the codes within this class. In particular, the dimension, the minimum distance, and the weight distribution of some ternary BCH codes with length <inline-formula> <tex-math notation="LaTeX">$n=(3^{m}-1)/2$ </tex-math></inline-formula> are determined in this paper. A class of ternary BCH codes meeting the Griesmer bound is identified. An application of some of the BCH codes in secret sharing is also investigated.

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