On Two Classes of Primitive BCH Codes and Some Related Codes

Chengju Li; Peng Wu; Fengmei Liu

doi:10.1109/TIT.2018.2883615

On Two Classes of Primitive BCH Codes and Some Related Codes

Chengju Li
, Peng Wu
, Fengmei Liu

Software Engineering Institute

Research output: Contribution to journal › Article › peer-review

36 Scopus citations

Abstract

BCH codes are an interesting type of cyclic codes and have wide applications in communication and storage systems. Generally, it is very hard to determine the minimum distances of BCH codes. In this paper, we determine the weight distributions of two classes of primitive BCH codes C _(q,m,δ2) and C _{(q, m, δ 3)} and their extended codes, which solve two problems proposed by Ding et al. It is shown that the extended codes C _(q,m,δ2) have four nonzero weights. We also employ the Hartmann-Tzeng bound to present the minimum distance of the dual code C _(q,m,δ2) perp for q ≥ 5. Inspired by the idea, we then determine the dimensions of a class of cyclic codes and give lower bounds on their minimum distances, which is greatly improved comparing with the BCH bound. Some optimal codes are obtained.

Original language	English
Article number	8550725
Pages (from-to)	3830-3840
Number of pages	11
Journal	IEEE Transactions on Information Theory
Volume	65
Issue number	6
DOIs	https://doi.org/10.1109/TIT.2018.2883615
State	Published - Jun 2019

Keywords

BCH codes
Hartmann-Tzeng bound
character sums
extended codes
weight distributions

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10.1109/TIT.2018.2883615

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title = "On Two Classes of Primitive BCH Codes and Some Related Codes",

abstract = " BCH codes are an interesting type of cyclic codes and have wide applications in communication and storage systems. Generally, it is very hard to determine the minimum distances of BCH codes. In this paper, we determine the weight distributions of two classes of primitive BCH codes C (q,m,δ2) and C (q, m, δ 3) and their extended codes, which solve two problems proposed by Ding et al. It is shown that the extended codes C (q,m,δ2) have four nonzero weights. We also employ the Hartmann-Tzeng bound to present the minimum distance of the dual code C (q,m,δ2) perp for q ≥ 5. Inspired by the idea, we then determine the dimensions of a class of cyclic codes and give lower bounds on their minimum distances, which is greatly improved comparing with the BCH bound. Some optimal codes are obtained.",

keywords = "BCH codes, Hartmann-Tzeng bound, character sums, extended codes, weight distributions",

author = "Chengju Li and Peng Wu and Fengmei Liu",

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AU - Wu, Peng

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AB - BCH codes are an interesting type of cyclic codes and have wide applications in communication and storage systems. Generally, it is very hard to determine the minimum distances of BCH codes. In this paper, we determine the weight distributions of two classes of primitive BCH codes C (q,m,δ2) and C (q, m, δ 3) and their extended codes, which solve two problems proposed by Ding et al. It is shown that the extended codes C (q,m,δ2) have four nonzero weights. We also employ the Hartmann-Tzeng bound to present the minimum distance of the dual code C (q,m,δ2) perp for q ≥ 5. Inspired by the idea, we then determine the dimensions of a class of cyclic codes and give lower bounds on their minimum distances, which is greatly improved comparing with the BCH bound. Some optimal codes are obtained.

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KW - Hartmann-Tzeng bound

KW - character sums

KW - extended codes

KW - weight distributions

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DO - 10.1109/TIT.2018.2883615

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On Two Classes of Primitive BCH Codes and Some Related Codes

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