A minimum distance bound for 2-dimension λ -quasi-twisted codes over finite fields

2018 ◽  
Vol 51 ◽  
pp. 146-167 ◽  
Author(s):  
Jingjie Lv ◽  
Jian Gao
Keyword(s):  
Finite Fields ◽  
Minimum Distance ◽  
Distance Bound ◽  
Minimum Distance Bound

scholarly journals From Generalization of Bacon-Shor Codes to High Performance Quantum LDPC Codes

2021 ◽  
Author(s):  
Jihao Fan ◽  
Jun Li ◽  
Ya Wang ◽  
Yonghui Li ◽  
Min-Hsiu Hsieh ◽  
...  
Keyword(s):  
Minimum Distance ◽  
High Performance ◽  
Ldpc Codes ◽  
Quantum Error Correction ◽  
Concatenated Codes ◽  
Quantum Codes ◽  
Error Correction Codes ◽  
Distance Bound ◽  
Quantum Error ◽  
Minimum Distance Bound

Abstract We utilize a concatenation scheme to construct new families of quantum error correction codes that include the Bacon-Shor codes. We show that our scheme can lead to asymptotically good quantum codes while Bacon-Shor codes cannot. Further, the concatenation scheme allows us to derive quantum LDPC codes of distance Ω(N2/3/loglogN) which can improve Hastings’s recent result [arXiv:2102.10030] by a polylogarithmic factor. Moreover, assisted by the Evra-Kaufman- Zémor distance balancing construction, our concatenation scheme can yield quantum LDPC codes with non-vanishing code rates and better minimum distance upper bound than the hypergraph product quantum LDPC codes. Finally, we derive a family of fast encodable and decodable quan- tum concatenated codes with parameters Q = [[N,Ω(√N),Ω(√N)]] and they also belong to the Bacon-Shor codes. We show that Q can be encoded very efficiently by circuits of size O(N) and depth O(√N), and can correct any adversarial error of weight up to half the minimum distance bound in O(√N) time. To the best of our knowledge, they are the most powerful quantum codes for correcting so many adversarial errors in sublinear time by far.

On some bounds on the minimum distance of cyclic codes over finite fields

2014 ◽  
Vol 76 (2) ◽  
pp. 173-178
Author(s):  
Ferruh Özbudak ◽  
Seher Tutdere ◽  
Oğuz Yayla
Keyword(s):  
Finite Fields ◽  
Minimum Distance ◽  
Cyclic Codes

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 Two classes of new optimal ternary cyclic codes

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Yan Liu ◽  
Xiwang Cao ◽  
Wei Lu
Keyword(s):  
Data Storage ◽  
Finite Fields ◽  
Weight Distribution ◽  
Communication Systems ◽  
Minimum Distance ◽  
Storage Systems ◽  
Cyclic Codes ◽  
Consumer Electronics

<p style='text-indent:20px;'>Due to their wide applications in consumer electronics, data storage systems and communication systems, cyclic codes have been an interesting subject of study in recent years. The construction of optimal cyclic codes over finite fields is important as they have maximal minimum distance once the length and dimension are given. In this paper, we present two classes of new optimal ternary cyclic codes <inline-formula><tex-math id="M1">\begin{document}$ \mathcal{C}_{(2,v)} $\end{document}</tex-math></inline-formula> by using monomials <inline-formula><tex-math id="M2">\begin{document}$ x^2 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ x^v $\end{document}</tex-math></inline-formula> for some suitable <inline-formula><tex-math id="M4">\begin{document}$ v $\end{document}</tex-math></inline-formula> and explain the novelty of the codes. Furthermore, the weight distribution of <inline-formula><tex-math id="M5">\begin{document}$ \mathcal{C}_{(2,v)}^{\perp} $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M6">\begin{document}$ v = \frac{3^{m}-1}{2}+2(3^{k}+1) $\end{document}</tex-math></inline-formula> is determined.</p>

scholarly journals Optimal quinary negacyclic codes with minimum distance four

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Jinmei Fan ◽  
Yanhai Zhang
Keyword(s):  
Finite Fields ◽  
Minimum Distance ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Negacyclic Codes ◽  
Generator Polynomial ◽  
Polynomials Over Finite Fields ◽  
Necessary And Sufficient

<p style='text-indent:20px;'>Based on solutions of certain equations over finite yields, a necessary and sufficient condition for the quinary negacyclic codes with parameters <inline-formula><tex-math id="M1">\begin{document}$ [\frac{5^m-1}{2},\frac{5^m-1}{2}-2m,4] $\end{document}</tex-math></inline-formula> to have generator polynomial <inline-formula><tex-math id="M2">\begin{document}$ m_{\alpha^3}(x)m_{\alpha^e}(x) $\end{document}</tex-math></inline-formula> is provided. Several classes of new optimal quinary negacyclic codes with the same parameters are constructed by analyzing irreducible factors of certain polynomials over finite fields. Moreover, several classes of new optimal quinary negacyclic codes with these parameters and generator polynomial <inline-formula><tex-math id="M3">\begin{document}$ m_{\alpha}(x)m_{\alpha^e}(x) $\end{document}</tex-math></inline-formula> are also presented.</p>

scholarly journals Minimum distance histograms with universal performance guarantees

2019 ◽  
Vol 2 (2) ◽  
pp. 507-527 ◽  
Author(s):  
Raazesh Sainudiin ◽  
Gloria Teng
Keyword(s):  
New York ◽  
Density Estimation ◽  
Minimum Distance ◽  
Estimation Methods ◽  
Real World Data ◽  
Performance Guarantees ◽  
Unknown Density ◽  
Distance Bound ◽  
Data Adaptive ◽  
Histogram Estimator

AbstractWe present a data-adaptive multivariate histogram estimator of an unknown density f based on n independent samples from it. Such histograms are based on binary trees called regular pavings (RPs). RPs represent a computationally convenient class of simple functions that remain closed under addition and scalar multiplication. Unlike other density estimation methods, including various regularization and Bayesian methods based on the likelihood, the minimum distance estimate (MDE) is guaranteed to be within an $$L_1$$ L 1 distance bound from f for a given n, no matter what the underlying f happens to be, and is thus said to have universal performance guarantees (Devroye and Lugosi, Combinatorial methods in density estimation. Springer, New York, 2001). Using a form of tree matrix arithmetic with RPs, we obtain the first generic constructions of an MDE, prove that it has universal performance guarantees and demonstrate its performance with simulated and real-world data. Our main contribution is a constructive implementation of an MDE histogram that can handle large multivariate data bursts using a tree-based partition that is computationally conducive to subsequent statistical operations.

Finite Fields

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