LCD Cyclic Codes Over Finite Fields

Li, Chengju; Ding, Cunsheng; Li, Shuxing

doi:10.1109/tit.2017.2672961

Cited by 175 publications

(139 citation statements)

References 18 publications

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“…Besides the results in Table I, for q tm{2u ă n ď q m´1 and 2 ď δ ď nq rm{2s q m´1 , the dimension of C pq,m,λ,δq was settled by Aly et al [3]. Recently, the dimensions of some BCH codes C pq,m,λ,δ,bq with b ‰ 0, 1 were settled in [28], [29], [35].…”

Section: B Known Resultsmentioning

confidence: 93%

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A Class of Narrow-Sense BCH Codes

Zhu

Sun

Kai

2019

IEEE Trans. Inform. Theory

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BCH codes are an important class of cyclic codes which have applications in satellite communications, DVDs, disk drives, and two-dimensional bar codes. Although BCH codes have been widely studied, their parameters are known for only a few special classes. Recently, Ding et al. made some new progress in BCH codes. However, we still have very limited knowledge on the dimension of BCH codes, not to mention the weight distribution of BCH codes. In this paper, we generalize the results on BCH codes from several previous papers.(i) The dimension of narrow-sense BCH codes of length q m´1 λ with designed distance 2 ď δ ď q rpm`1q{2s´1 λ`1 is settled, where λ is any factor of q´1.(ii) The weight distributions of two classes of narrow-sense BCH codes of length q m´1 2 with designed distance δ " pq´1qq m´1´qtpm´1q{2u´1 2 and δ " pq´1qq m´1´qtpm`1q{2u´1 2 are determined. (iii) The weight distribution of a class of BCH codes of length q m´1 q´1 is determined. In particular, a subclass of this class of BCH codes is optimal with respect to the Griesmer bound. Some optimal linear codes obtained from this class of BCH codes are characterized.

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Section: B Known Resultsmentioning

confidence: 93%

“…[48] q m{2`2 ď δ ď 2q m{2`1 , m even 2 ď δ ď q rm{2s`1 [29] λ " q´1 2 ď δ ď q m{2 , m even [28] 2 ď δ ď q pm`1q{2 , m odd [29] λ " q ℓ´1 , m " 2ℓ 3 ď δ ď q tpℓ´1q{2u`2 [28] 2 ď δ ď q tpℓ`1q{2u`1 [29] 2 ď δ ď 2q ℓ{2`3 , ℓ even; [33] 2 ď δ ď 2q pℓ`1q{2`2 q, ℓ odd.…”

Section: A Backgroundsunclassified

A Class of Narrow-Sense BCH Codes

Zhu

Sun

Kai

2019

IEEE Trans. Inform. Theory

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“…A cyclic code C is called reversible if a ∈ C implies a r ∈ C. A reversible cyclic code is also known as an LCD cyclic code and has been studied extensively [6], [10]- [12]. In this paper, reversible cyclic codes containing the all-one vector 1 n are of particular interest.…”

Section: A Cyclic and Reversible Codesmentioning

confidence: 99%

Efficient and Explicit Balanced Primer Codes

Chee

Kiah

Wei

2019

2019 IEEE International Symposium on Information Theory (ISIT)

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To equip DNA-based data storage with random-access capabilities, Yazdi et al. (2018) prepended DNA strands with specially chosen address sequences called primers and provided certain design criteria for these primers. We provide explicit constructions of errorcorrecting codes that are suitable as primer addresses and equip these constructions with efficient encoding algorithms.Specifically, our constructions take cyclic or linear codes as inputs and produce sets of primers with similar error-correcting capabilities. Using certain classes of BCH codes, we obtain infinite families of primer sets of length n, minimum distance d with (d + 1) log 4 n + O(1) redundant symbols. Our techniques involve reversible cyclic codes (1964), an encoding method of Tavares et al. (1971) and Knuth's balancing technique (1986). In our investigation, we also construct efficient and explicit binary balanced errorcorrecting codes and codes for DNA computing.

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“…A practical application of binary LCDs against sidechannel attacks (SCAs) and fault injection attacks (FIAs) was investigated by Carlet et al in [10] and [11]. The study of LCD codes is thus becoming a hot research topic in coding theory ( [12,13,14,15,16]). A surprising result was given in [16], which proved that any linear code over F q (q > 3) is equivalent to a Euclidean LCD code and any linear code over F q 2 (q > 2) is equivalent to a Hermitian LCD code.…”

Section: Introductionmentioning

confidence: 99%

Euclidean and Hermitian Hulls of MDS Codes and Their Applications to EAQECCs

Fang

et al. 2020

IEEE Trans. Inform. Theory

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In this paper, we construct several classes of maximum distance separable (MDS) codes via generalized Reed-Solomon (GRS) codes and extended GRS codes, which we can determine the dimensions of their Euclidean hulls or Hermitian hulls. It turns out that the dimensions of Euclidean hulls or Hermitian hulls of the codes in our constructions can take all or almost all possible values. As a consequence, we can apply our results to entanglement-assisted quantum error-correcting codes (EAQECCs) and obtain several families of MDS EAQECCs with flexible parameters. The required number of maximally entangled states of these MDS EAQECCs can take all or almost all possible values. In particular, all possible parameters for the q-ary MDS EAQECCs of length n ≤ q are completely determined and several new classes of q-ary MDS EAQECCs of length n > q are also obtained.

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LCD Cyclic Codes Over Finite Fields

Cited by 175 publications

References 18 publications

A Class of Narrow-Sense BCH Codes

A Class of Narrow-Sense BCH Codes

Efficient and Explicit Balanced Primer Codes

Euclidean and Hermitian Hulls of MDS Codes and Their Applications to EAQECCs

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