scholarly journals On the weight distribution of the cosets of MDS codes

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Alexander A. Davydov ◽  
Stefano Marcugini ◽  
Fernanda Pambianco
Keyword(s):  
Weight Distribution ◽  
Minimum Distance ◽  
Covering Radius ◽  
Mds Codes ◽  
Mds Code ◽  
Weight Distributions ◽  
Maximum Distance Separable ◽  
Special Relations ◽  
Saturating Set

<p style='text-indent:20px;'>The weight distribution of the cosets of maximum distance separable (MDS) codes is considered. In 1990, P.G. Bonneau proposed a relation to obtain the full weight distribution of a coset of an MDS code with minimum distance <inline-formula><tex-math id="M1">\begin{document}$ d $\end{document}</tex-math></inline-formula> using the known numbers of vectors of weights <inline-formula><tex-math id="M2">\begin{document}$ \le d-2 $\end{document}</tex-math></inline-formula> in this coset. In this paper, the Bonneau formula is transformed into a more structured and convenient form. The new version of the formula allows to consider effectively cosets of distinct weights <inline-formula><tex-math id="M3">\begin{document}$ W $\end{document}</tex-math></inline-formula>. (The weight <inline-formula><tex-math id="M4">\begin{document}$ W $\end{document}</tex-math></inline-formula> of a coset is the smallest Hamming weight of any vector in the coset.) For each of the considered <inline-formula><tex-math id="M5">\begin{document}$ W $\end{document}</tex-math></inline-formula> or regions of <inline-formula><tex-math id="M6">\begin{document}$ W $\end{document}</tex-math></inline-formula>, special relations more simple than the general ones are obtained. For the MDS code cosets of weight <inline-formula><tex-math id="M7">\begin{document}$ W = 1 $\end{document}</tex-math></inline-formula> and weight <inline-formula><tex-math id="M8">\begin{document}$ W = d-1 $\end{document}</tex-math></inline-formula> we obtain formulas of the weight distributions depending only on the code parameters. This proves that all the cosets of weight <inline-formula><tex-math id="M9">\begin{document}$ W = 1 $\end{document}</tex-math></inline-formula> (as well as <inline-formula><tex-math id="M10">\begin{document}$ W = d-1 $\end{document}</tex-math></inline-formula>) have the same weight distribution. The cosets of weight <inline-formula><tex-math id="M11">\begin{document}$ W = 2 $\end{document}</tex-math></inline-formula> or <inline-formula><tex-math id="M12">\begin{document}$ W = d-2 $\end{document}</tex-math></inline-formula> may have different weight distributions; in this case, we proved that the distributions are symmetrical in some sense. The weight distributions of the cosets of MDS codes corresponding to arcs in the projective plane <inline-formula><tex-math id="M13">\begin{document}$ \mathrm{PG}(2,q) $\end{document}</tex-math></inline-formula> are also considered. For MDS codes of covering radius <inline-formula><tex-math id="M14">\begin{document}$ R = d-1 $\end{document}</tex-math></inline-formula> we obtain the number of the weight <inline-formula><tex-math id="M15">\begin{document}$ W = d-1 $\end{document}</tex-math></inline-formula> cosets and their weight distribution that gives rise to a certain classification of the so-called deep holes. We show that any MDS code of covering radius <inline-formula><tex-math id="M16">\begin{document}$ R = d-1 $\end{document}</tex-math></inline-formula> is an almost perfect multiple covering of the farthest-off points (deep holes); moreover, it corresponds to an optimal multiple saturating set in the projective space <inline-formula><tex-math id="M17">\begin{document}$ \mathrm{PG}(N,q) $\end{document}</tex-math></inline-formula>.</p>

scholarly journals ON OPTIMAL QUANTUM CODES

2004 ◽  
Vol 02 (01) ◽  
pp. 55-64 ◽  
Author(s):  
MARKUS GRASSL ◽  
THOMAS BETH ◽  
MARTIN RÖTTELER
Keyword(s):  
Minimum Distance ◽  
Prime Power ◽  
Quantum Systems ◽  
Error Correcting Codes ◽  
Mds Codes ◽  
Quantum Codes ◽  
Maximum Distance Separable ◽  
Shortened Codes ◽  
Quantum Error ◽  
Quantum Mds Codes

We present families of quantum error-correcting codes which are optimal in the sense that the minimum distance is maximal. These maximum distance separable (MDS) codes are defined over q-dimensional quantum systems, where q is an arbitrary prime power. It is shown that codes with parameters 〚n, n - 2d + 2, d〛q exist for all 3≤n≤q and 1≤d≤n/2+1. We also present quantum MDS codes with parameters 〚q2, q2-2d+2, d〛q for 1≤d≤q which additionally give rise to shortened codes 〚q2-s, q2-2d+2-s, d〛q for some s.

New entanglement-assisted quantum MDS codes

2021 ◽  
pp. 2150016
Author(s):  
Binbin Pang ◽  
Shixin Zhu ◽  
Liqi Wang
Keyword(s):  
Coding Theory ◽  
Minimum Distance ◽  
Linear Codes ◽  
Error Correcting Codes ◽  
Mds Codes ◽  
Maximum Distance Separable ◽  
Number Formula ◽  
Minimum Distances ◽  
Quantum Error ◽  
Quantum Mds Codes

Entanglement-assisted quantum error-correcting codes (EAQECCs) can be obtained from arbitrary classical linear codes based on the entanglement-assisted stabilizer formalism, which greatly promoted the development of quantum coding theory. In this paper, we construct several families of [Formula: see text]-ary entanglement-assisted quantum maximum-distance-separable (EAQMDS) codes of lengths [Formula: see text] with flexible parameters as to the minimum distance [Formula: see text] and the number [Formula: see text] of maximally entangled states. Most of the obtained EAQMDS codes have larger minimum distances than the codes available in the literature.

New constructions of entanglement-assisted quantum MDS codes from negacyclic codes

2019 ◽  
Vol 17 (03) ◽  
pp. 1950022 ◽  
Author(s):  
Ruihu Li ◽  
Guanmin Guo ◽  
Hao Song ◽  
Yang Liu
Keyword(s):  
Minimum Distance ◽  
Linear Codes ◽  
Error Correcting Codes ◽  
Optimal Number ◽  
Mds Codes ◽  
Quantum Codes ◽  
General Statement ◽  
Negacyclic Codes ◽  
Maximum Distance Separable ◽  
Quantum Error

When constructing quantum codes under the entanglement-assisted (EA) stabilizer formalism, one can ignore the limitation of dual-containing condition. This allows us to construct EA quantum error-correcting codes (QECCs) from any classical linear codes. The main contribution of this manuscript is to make a general statement for determining the optimal number of pre-shared qubits instead of presenting only specific cases. Let [Formula: see text] and [Formula: see text], where [Formula: see text] is an odd prime power, [Formula: see text] and [Formula: see text]. By deeply investigating the decomposition of the defining set of negacyclic codes, we generalize the number of pre-shared entanglement pairs of Construction (1) in Lu et al. [Quantom Inf. Process. 17 (2018) 69] from [Formula: see text] to arbitrary even numbers less than or equal to [Formula: see text]. Consequently, a series of EA quantum maximum distance separable (EAQMDS) codes can be produced. The absolute majority of them are new and the minimum distance can be up to [Formula: see text]. Moreover, this method can be applied to construct many other families of EAQECCs with good parameters, especially large minimum distance.

Construction of new quantum MDS codes derived from constacyclic codes

2017 ◽  
Vol 15 (01) ◽  
pp. 1750008 ◽  
Author(s):  
Divya Taneja ◽  
Manish Gupta ◽  
Rajesh Narula ◽  
Jaskaran Bhullar
Keyword(s):  
Minimum Distance ◽  
Maximum Distance ◽  
Mds Codes ◽  
Constacyclic Codes ◽  
Maximum Distance Separable ◽  
Work Done ◽  
Quantum Mds Codes

Obtaining quantum maximum distance separable (MDS) codes from dual containing classical constacyclic codes using Hermitian construction have paved a path to undertake the challenges related to such constructions. Using the same technique, some new parameters of quantum MDS codes have been constructed here. One set of parameters obtained in this paper has achieved much larger distance than work done earlier. The remaining constructed parameters of quantum MDS codes have large minimum distance and were not explored yet.

New Results on Codes with Covering Radius 1 and Minimum Distance 2

2005 ◽  
Vol 35 (2) ◽  
pp. 241-250 ◽  
Author(s):  
Patric R. J. �sterg�rd ◽  
J�rn Quistorff ◽  
Alfred Wassermann
Keyword(s):  
Minimum Distance ◽  
Covering Radius

scholarly journals On Mixed Codes with Covering Radius $1$ and Minimum Distance $2$

10.37236/969 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Wolfgang Haas ◽  
Jörn Quistorff
Keyword(s):  
Lower Bounds ◽  
Minimum Distance ◽  
Covering Radius ◽  
Minimal Cardinality ◽  
Finite Sets ◽  
Open Problems ◽  
Latin Rectangle ◽  
Mixed Codes ◽  
Special Case

Let $R$, $S$ and $T$ be finite sets with $|R|=r$, $|S|=s$ and $|T|=t$. A code $C\subset R\times S\times T$ with covering radius $1$ and minimum distance $2$ is closely connected to a certain generalized partial Latin rectangle. We present various constructions of such codes and some lower bounds on their minimal cardinality $K(r,s,t;2)$. These bounds turn out to be best possible in many instances. Focussing on the special case $t=s$ we determine $K(r,s,s;2)$ when $r$ divides $s$, when $r=s-1$, when $s$ is large, relative to $r$, when $r$ is large, relative to $s$, as well as $K(3r,2r,2r;2)$. Some open problems are posed. Finally, a table with bounds on $K(r,s,s;2)$ is given.

Characterization of Elution Chromatography Fractions from CIS-Polybutadiene by GPC

1970 ◽  
Vol 43 (6) ◽  
pp. 1439-1450 ◽  
Author(s):  
W. V. Smith ◽  
S. Thiruvengada
Keyword(s):  
Molecular Weight ◽  
Molecular Weight Distribution ◽  
Weight Distribution ◽  
Low Molecular Weight ◽  
Molecular Weight Distributions ◽  
Weight Distributions ◽  
Log Normal ◽  
Analytical Fractionation ◽  
Molecular Weight Fractions ◽  
Elution Chromatography

Abstract A preparative fractionation of about 23 g of a commercial cis-polybutadiene rubber is described. The method employed was a solvent elution chromatographic method with very little temperature gradient. The molecular weight distributions of the fractions obtained were determined by an analytical fractionation of 20 mg of polymer. The method was similar to the preparative fractionation and involved solvent elution chromatography. The fractions obtained were assayed for quantity, molecular weight, and molecular weight distribution by GPC. The low molecular weight fractions of the preparative fractionation had molecular weight distributions which could be closely approximated by two log normal distributions, the low molecular weight component having the narrower width. The ratio of weight to number average molecular weight was found to be about 1.1 for these samples. The higher molecular weight fractions could also be approximated by two log normal distributions; however, in these fractions the low molecular weight component had a very broad distribution but constituted only a small portion of the sample. The widths of the GPC curves of the fractions correlate satisfactorily with the molecular weight distributions found by the analytical refractionations. The GPC width is a sensitive criterion of the width of the molecular weight distribution even when only two columns are used. It is felt that the analytical fractionation procedure presented gives more detailed information on the molecular weight distribution than is easily obtainable from an ordinary GPC curve.

Constructing new q-ary quantum MDS codes with distances bigger than q/2 from generator matrices

2018 ◽  
Vol 18 (3&4) ◽  
pp. 223-230
Author(s):  
Xianmang He
Keyword(s):  
Information Theory ◽  
Quantum Information ◽  
Minimum Distance ◽  
Quantum Information Theory ◽  
Error Correcting Codes ◽  
Mds Codes ◽  
Quantum Error ◽  
Generator Matrices ◽  
Quantum Mds Codes

The construction of quantum error-correcting codes has been an active field of quantum information theory since the publication of \cite{Shor1995Scheme,Steane1998Enlargement,Laflamme1996Perfect}. It is becoming more and more difficult to construct some new quantum MDS codes with large minimum distance. In this paper, based on the approach developed in the paper \cite{NewHeMDS2016}, we construct several new classes of quantum MDS codes. The quantum MDS codes exhibited here have not been constructed before and the distance parameters are bigger than q/2.

scholarly journals Tuning the molecular weight distribution from atom transfer radical polymerization using deep reinforcement learning

2018 ◽  
Vol 3 (3) ◽  
pp. 496-508 ◽  
Author(s):  
Haichen Li ◽  
Christopher R. Collins ◽  
Thomas G. Ribelli ◽  
Krzysztof Matyjaszewski ◽  
Geoffrey J. Gordon ◽  
...  
Keyword(s):  
Molecular Weight ◽  
Reinforcement Learning ◽  
Atom Transfer Radical Polymerization ◽  
Radical Polymerization ◽  
Weight Distribution ◽  
In Silico ◽  
Atom Transfer ◽  
Polymer Molecular Weight ◽  
Molecular Weight Distributions ◽  
Weight Distributions

Combination of deep reinforcement learning and atom transfer radical polymerization gives precise in silico control on polymer molecular weight distributions.

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