scholarly journals Rank-deficient representations in the theta correspondence over finite fields arise from quantum codes

2021 ◽  
Vol 25 (8) ◽  
pp. 193-223
Author(s):  
Felipe Montealegre-Mora ◽  
David Gross
Keyword(s):  
Finite Fields ◽  
Quantum Codes ◽  
Theta Correspondence

scholarly journals AG codes from $${{\mathbb{F}}_{q^7}}$$-rational points of the GK maximal curve

2021 ◽  
Author(s):  
Stefano Lia ◽  
Marco Timpanella
Keyword(s):  
Finite Fields ◽  
Minimum Distance ◽  
Rational Points ◽  
Quantum Codes ◽  
Ag Codes ◽  
Maximal Curve ◽  
Weierstrass Semigroups ◽  
Minimal Generators

AbstractIn Beelen and Montanucci (Finite Fields Appl 52:10–29, 2018) and Giulietti and Korchmáros (Math Ann 343:229–245, 2009), Weierstrass semigroups at points of the Giulietti–Korchmáros curve $${\mathcal {X}}$$ X were investigated and the sets of minimal generators were determined for all points in $${\mathcal {X}}(\mathbb {F}_{q^2})$$ X ( F q 2 ) and $${\mathcal {X}}(\mathbb {F}_{q^6})\setminus {\mathcal {X}}( \mathbb {F}_{q^2})$$ X ( F q 6 ) \ X ( F q 2 ) . This paper completes their work by settling the remaining cases, that is, for points in $${\mathcal {X}}(\overline{\mathbb {F}}_{q}){\setminus }{\mathcal {X}}( \mathbb {F}_{q^6})$$ X ( F ¯ q ) \ X ( F q 6 ) . As an application to AG codes, we determine the dimensions and the lengths of duals of one-point codes from a point in $${\mathcal {X}}(\mathbb {F}_{q^7}){\setminus }{\mathcal {X}}( \mathbb {F}_{q})$$ X ( F q 7 ) \ X ( F q ) and we give a bound on the Feng–Rao minimum distance $$d_{ORD}$$ d ORD . For $$q=3$$ q = 3 we provide a table that also reports the exact values of $$d_{ORD}$$ d ORD . As a further application we construct quantum codes from $$\mathbb {F}_{q^7}$$ F q 7 -rational points of the GK-curve.

scholarly journals Remarks on the theta correspondence over finite fields

2020 ◽  
Vol 306 (2) ◽  
pp. 587-609
Author(s):  
Dongwen Liu ◽  
Zhicheng Wang
Keyword(s):  
Finite Fields ◽  
Theta Correspondence

scholarly journals Some New Quantum BCH Codes over Finite Fields

Entropy ◽  
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.

Finite Fields

1996 ◽  
Author(s):  
Rudolf Lidl ◽  
Harald Niederreiter
Keyword(s):  
Finite Fields

Finite Fields and Applications

1996 ◽  
Keyword(s):  
Finite Fields

Algebraic Curves over Finite Fields

1991 ◽  
Author(s):  
Carlos Moreno
Keyword(s):  
Finite Fields ◽  
Algebraic Curves ◽  
Curves Over Finite Fields

Maximum Distance Separable Hankel Rhotrices over Finite Fields

2018 ◽  
Vol 43 (1-4) ◽  
pp. 13-45
Author(s):  
Prof. P. L. Sharma ◽  
◽  
Mr. Arun Kumar ◽  
Mrs. Shalini Gupta ◽  
◽  
...  
Keyword(s):  
Finite Fields ◽  
Maximum Distance ◽  
Maximum Distance Separable

Notes on three-pass protocol

2020 ◽  
Vol 25 (4) ◽  
pp. 4-9
Author(s):  
Yerzhan R. Baissalov ◽  
Ulan Dauyl
Keyword(s):  
Finite Fields ◽  
Matrix Algebras ◽  
Associative Structures

The article discusses primitive, linear three-pass protocols, as well as three-pass protocols on associative structures. The linear three-pass protocols over finite fields and the three-pass protocols based on matrix algebras are shown to be cryptographically weak.

About the Complexity of Square Root Extraction Algorithms in Finite Fields and Residue Rings

Vestnik MEI ◽  
2018 ◽  
Vol 5 (5) ◽  
pp. 79-88
Author(s):  
Sergey B. Gashkov ◽  
◽  
Aleksandr B. Frolov ◽  
Elizaveta Р. Popova ◽  
◽  
...  
Keyword(s):  
Finite Fields ◽  
Square Root ◽  
Root Extraction
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