Finite fields and BCH codes

This paper overviews the study of skewΘ-λ-constacyclic codes over finite fields and finite commutative chain rings. The structure of skewΘ-λ-constacyclic codes and their duals are provided. Among other results, we also consider the Euclidean and Hermitian dual codes of skewΘ-cyclic and skewΘ-negacyclic codes over finite chain rings in general and overFpm+uFpmin particular. Moreover, general decoding procedure for decoding skew BCH codes with designed distance and an algorithm for decoding skew BCH codes are discussed.

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Designed distances and parameters of new LCD BCH codes over finite fields

Cryptography and Communications ◽

10.1007/s12095-019-00385-3 ◽

2019 ◽

Vol 12 (1) ◽

pp. 147-163 ◽

Cited By ~ 1

Author(s):

Fengwei Li ◽

Qin Yue ◽

Yansheng Wu

Keyword(s):

Finite Fields ◽

Bch Codes

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A Flexible Hybrid BCH Decoder for Modern NAND Flash Memories Using General Purpose Graphical Processing Units (GPGPUs)

Micromachines ◽

10.3390/mi10060365 ◽

2019 ◽

Vol 10 (6) ◽

pp. 365 ◽

Cited By ~ 4

Author(s):

Arul Subbiah ◽

Tokunbo Ogunfunmi

Keyword(s):

Finite Fields ◽

Flash Memory ◽

Search Algorithm ◽

Memory Systems ◽

Block Codes ◽

General Purpose ◽

Bch Codes ◽

Nand Flash ◽

Graphical Processing Units ◽

Graphical Processing

Bose–Chaudhuri–Hocquenghem (BCH) codes are broadly used to correct errors in flash memory systems and digital communications. These codes are cyclic block codes and have their arithmetic fixed over the splitting field of their generator polynomial. There are many solutions proposed using CPUs, hardware, and Graphical Processing Units (GPUs) for the BCH decoders. The performance of these BCH decoders is of ultimate importance for systems involving flash memory. However, it is essential to have a flexible solution to correct multiple bit errors over the different finite fields (GF(2 m )). In this paper, we propose a pragmatic approach to decode BCH codes over the different finite fields using hardware circuits and GPUs in tandem. We propose to employ hardware design for a modified syndrome generator and GPUs for a key-equation solver and an error corrector. Using the above partition, we have shown the ability to support multiple bit errors across different BCH block codes without compromising on the performance. Furthermore, the proposed method to generate modified syndrome has zero latency for scenarios where there are no errors. When there is an error detected, the GPUs are deployed to correct the errors using the iBM and Chien search algorithm. The results have shown that using the modified syndrome approach, we can support different multiple finite fields with high throughput.

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Some New Quantum BCH Codes over Finite Fields

Entropy ◽

10.3390/e23060712 ◽

2021 ◽

Vol 23 (6) ◽

pp. 712

Author(s):

Lijuan Xing ◽

Zhuo Li

Keyword(s):

Finite Fields ◽

Error Correcting Codes ◽

Bch Codes ◽

Quantum Codes ◽

Dual Codes ◽

Cyclotomic Cosets ◽

New Family ◽

Consecutive Integers ◽

Css Construction ◽

Quantum Error

Quantum error correcting codes (QECCs) play an important role in preventing quantum information decoherence. Good quantum stabilizer codes were constructed by classical error correcting codes. In this paper, Bose–Chaudhuri–Hocquenghem (BCH) codes over finite fields are used to construct quantum codes. First, we try to find such classical BCH codes, which contain their dual codes, by studying the suitable cyclotomic cosets. Then, we construct nonbinary quantum BCH codes with given parameter sets. Finally, a new family of quantum BCH codes can be realized by Steane’s enlargement of nonbinary Calderbank-Shor-Steane (CSS) construction and Hermitian construction. We have proven that the cyclotomic cosets are good tools to study quantum BCH codes. The defining sets contain the highest numbers of consecutive integers. Compared with the results in the references, the new quantum BCH codes have better code parameters without restrictions and better lower bounds on minimum distances. What is more, the new quantum codes can be constructed over any finite fields, which enlarges the range of quantum BCH codes.

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