The dimension and minimum distance of two classes of primitive BCH codes

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Abstract

Cyclic Reed–Solomon codes, a type of BCH codes, are widely used in consumer electronics, communication systems, and data storage devices. This fact demonstrates the importance of BCH codes – a family of cyclic codes – in practice. In theory, BCH codes are among the best cyclic codes in terms of their error-correcting capability. A subclass of BCH codes are the narrow-sense primitive BCH codes. However, the dimension and minimum distance of these codes are not known in general. The objective of this paper is to determine the dimension and minimum distances of two classes of narrow-sense primitive BCH codes with designed distances δ=(q1)qm11q(m1)/2 and δ=(q1)qm11q(m+1)/2. The weight distributions of some of these BCH codes are also reported. As will be seen, the two classes of BCH codes are sometimes optimal and sometimes among the best linear codes known.

MSC

94B15
11T71

Keywords

BCH codes
Cyclic codes
Linear codes
Secret sharing
Weight distribution
Weight enumerator

Cited by (0)

C. Ding's research was supported by the Hong Kong Research Grants Council, Proj. No. 16300415. C. Fan and Z. Zhou were supported by the Natural Science Foundation of China under Grants 11571285 and 61672028, and also the Sichuan Provincial Youth Science and Technology Fund under Grant 2015JQ0004 and 2016JQ0004. C. Fan is also with State Key Laboratory of Information Security (Institute of Information Engineering, Chinese Academy of Sciences, Beijing 100093).