scholarly journals Quantum stabilizer codes from Abelian and non-Abelian groups association schemes

2015 ◽  
Vol 13 (03) ◽  
pp. 1550021 ◽  
Author(s):  
Avaz Naghipour ◽  
Mohammad Ali Jafarizadeh ◽  
Sedaghat Shahmorad
Keyword(s):  
Abelian Group ◽  
Abelian Groups ◽  
Regular Representation ◽  
Association Schemes ◽  
Stabilizer Codes ◽  
Cyclic Groups ◽  
Minimum Distances ◽  
Quantum Stabilizer Code ◽  
Group Association ◽  
Quantum Stabilizer Codes

A new method for the construction of the binary quantum stabilizer codes is provided, where the construction is based on Abelian and non-Abelian groups association schemes. The association schemes based on non-Abelian groups are constructed by bases for the regular representation from U6n, T4n, V8n and dihedral D2n groups. By using Abelian group association schemes followed by cyclic groups and non-Abelian group association schemes a list of binary stabilizer codes up to 40 qubits is given in tables 4, 5 and 10. Moreover, several binary stabilizer codes of minimum distances 5, 7 and 8 with good quantum parameters is presented. The preference of this method specially for Abelian group association schemes is that one can construct any binary quantum stabilizer code with any distance by using the commutative structure of association schemes.

Criteria for Total Projectivity

1981 ◽  
Vol 33 (4) ◽  
pp. 817-825 ◽  
Author(s):  
Paul Hill
Keyword(s):  
Abelian Group ◽  
Positive Integer ◽  
Abelian Groups ◽  
General Criterion ◽  
Direct Sums ◽  
Primary Abelian Group ◽  
Cyclic Groups ◽  
Direct Sum ◽  
Totally Projective Groups ◽  
Torsion Groups

All groups herein are assumed to be abelian. It was not until the 1940's that it was known that a subgroup of an infinite direct sum of finite cyclic groups is again a direct sum of cyclics. This result rests on a general criterion due to Kulikov [7] for a primary abelian group to be a direct sum of cyclic groups. If G is p-primary, Kulikov's criterion presupposes that G has no elements (other than zero) having infinite p-height. For such a group G, the criterion is simply that G be the union of an ascending sequence of subgroups Hn where the heights of the elements of Hn computed in G are bounded by some positive integer λ(n). The theory of abelian groups has now developed to the point that totally projective groups currently play much the same role, at least in the theory of torsion groups, that direct sums of cyclic groups and countable groups played in combination prior to the discovery of totally projective groups and their structure beginning with a paper by R. Nunke [11] in 1967.

Graphical Regular Representations of Non-Abelian Groups, II

1972 ◽  
Vol 24 (6) ◽  
pp. 1009-1018 ◽  
Author(s):  
Lewis A. Nowitz ◽  
Mark E. Watkins
Keyword(s):  
Abelian Group ◽  
Abelian Groups ◽  
Regular Representation ◽  
Affirmative Answer ◽  
Roman Numeral ◽  
Regular Representations ◽  
Graphical Regular Representation ◽  
Bold Face

The present paper is a sequel to the previous paper bearing the same title by the same authors [3] and which will be hereafter designated by the bold-face Roman numeral I. Further results are obtained in determining whether a given finite non-abelian group G has a graphical regular representation. In particular, an affirmative answer will be given if (|G|, 6) = 1.Inasmuch as much of the machinery of I will be required in the proofs to be presented and a perusal of I is strongly recommended to set the stage and provide motivation for this paper, an independent and redundant introduction will be omitted in the interest of economy.

scholarly journals New Constructions of Quantum Stabilizer Codes Based on Difference Sets

Symmetry ◽  
2018 ◽  
Vol 10 (11) ◽  
pp. 655 ◽  
Author(s):  
Duc Nguyen ◽  
Sunghwan Kim
Keyword(s):  
Difference Sets ◽  
Combinatorial Design ◽  
Inner Product ◽  
Code Construction ◽  
Stabilizer Codes ◽  
Stabilizer Code ◽  
The Difference ◽  
Symplectic Inner Product ◽  
Quantum Stabilizer Code ◽  
Quantum Stabilizer Codes

In this paper, new conditions on parameters in difference sets are derived to satisfy symplectic inner product, and new constructions of quantum stabilizer codes are proposed from the conditions. The conversion of the difference sets into parity-check matrices is first explained. Then, the proposed code construction is composed of three steps, which are to choose the generators of quantum stabilizer code, to determine the quantum stabilizer groups, and to determine subspace codewords with large minimum distance. The quantum stabilizer codes with various length are also presented to explain the practicality of the code construction. The proposed design can be applied to quantum stabilizer code construction based on combinatorial design.

scholarly journals The Number of Subgroup Chains of Finite Nilpotent Groups

Symmetry ◽  
2020 ◽  
Vol 12 (9) ◽  
pp. 1537 ◽  
Author(s):  
Lingling Han ◽  
Xiuyun Guo
Keyword(s):  
Abelian Group ◽  
Abelian Groups ◽  
Finite Abelian Group ◽  
Recursive Formula ◽  
Nilpotent Groups ◽  
Classification Problem ◽  
Finite Abelian Groups ◽  
Counting Problem ◽  
Cyclic Groups ◽  
Fuzzy Subgroups

In this paper, we mainly count the number of subgroup chains of a finite nilpotent group. We derive a recursive formula that reduces the counting problem to that of finite p-groups. As applications of our main result, the classification problem of distinct fuzzy subgroups of finite abelian groups is reduced to that of finite abelian p-groups. In particular, an explicit recursive formula for the number of distinct fuzzy subgroups of a finite abelian group whose Sylow subgroups are cyclic groups or elementary abelian groups is given.

A novel construction for quantum stabilizer codes based on binary formalism

2020 ◽  
Vol 34 (08) ◽  
pp. 2050059 ◽  
Author(s):  
Duc Manh Nguyen ◽  
Sunghwan Kim
Keyword(s):  
Minimum Distance ◽  
Binary Vector ◽  
Quantum Codes ◽  
Quantum Code ◽  
Parity Check Matrix ◽  
Stabilizer Codes ◽  
Check Matrix ◽  
Stabilizer Code ◽  
Quantum Stabilizer Code ◽  
Quantum Stabilizer Codes

In this research, we propose a novel construction of quantum stabilizer code based on a binary formalism. First, from any binary vector of even length, we generate the parity-check matrix of the quantum code from a set composed of elements from this vector and its relations by shifts via subtraction and addition. We prove that the proposed matrices satisfy the condition constraint for the construction of quantum codes. Finally, we consider some constraint vectors which give us quantum stabilizer codes with various dimensions and a large minimum distance with code length from six to twelve digits.

scholarly journals Rainbow-free $3$-colorings of Abelian Groups

10.37236/2054 ◽  
2012 ◽  
Vol 19 (1) ◽  
Author(s):  
Amanda Montejano ◽  
Oriol Serra
Keyword(s):  
Abelian Group ◽  
Arithmetic Progression ◽  
Structural Characterization ◽  
Abelian Groups ◽  
Cyclic Groups ◽  
Chromatic Class

A $3$-coloring of the elements of an abelian group is said to be rainbow-free if there is no $3$-term arithmetic progression with its members having pairwise distinct colors. We give a structural characterization of rainbow-free colorings of abelian groups. This characterization proves a conjecture of Jungić et al. on the size of the smallest chromatic class of a rainbow-free $3$-coloring of cyclic groups.

On the Splitting of Modules and Abelian Groups

1974 ◽  
Vol 26 (1) ◽  
pp. 68-77 ◽  
Author(s):  
Paul Hill
Keyword(s):  
Abelian Group ◽  
Abelian Groups ◽  
Cyclic Groups ◽  
Countable Subgroup ◽  
Free Abelian Groups ◽  
Torsion Free ◽  
Infinite Abelian Group ◽  
Fundamental Paper

In a fundamental paper on torsion-free abelian groups, R. Baer [1] proved that the group P of all sequences of integers with respect to componentwise addition is not free. This means precisely that P is not a direct sum of infinite cyclic groups. However, E. Specker proved in [9] that P has the property that any countable subgroup is free. Since an infinite abelian group G is called -free if each subgroup of rank less than is free, these results are equivalent to: P is -free but not free.

On Restricted Sums

2000 ◽  
Vol 9 (6) ◽  
pp. 513-518 ◽  
Author(s):  
Y. O. HAMIDOUNE ◽  
A. S. LLADÓ ◽  
O. SERRA
Keyword(s):  
Abelian Group ◽  
Prime Order ◽  
Abelian Groups ◽  
Generating Set ◽  
Cyclic Groups ◽  
Odd Order

Let G be an abelian group. For a subset A ⊂ G, denote by 2 ∧ A the set of sums of two different elements of A. A conjecture by Erdős and Heilbronn, first proved by Dias da Silva and Hamidoune, states that, when G has prime order, [mid ]2 ∧ A[mid ] [ges ] min([mid ]G[mid ], 2[mid ]A[mid ] − 3).We prove that, for abelian groups of odd order (respectively, cyclic groups), the inequality [mid ]2 ∧ A[mid ] [ges ] min([mid ]G[mid ], 3[mid ]A[mid ]/2) holds when A is a generating set of G, 0 ∈ A and [mid ]A[mid ] [ges ] 21 (respectively, [mid ]A[mid ] [ges ] 33). The structure of the sets for which equality holds is also determined.

THE MAXIMUM SIZE OF -SUM-FREE SETS IN CYCLIC GROUPS

2018 ◽  
Vol 99 (2) ◽  
pp. 184-194
Author(s):  
BÉLA BAJNOK ◽  
RYAN MATZKE
Keyword(s):  
Number Theory ◽  
Abelian Group ◽  
Explicit Formula ◽  
Abelian Groups ◽  
Finite Abelian Group ◽  
Maximum Size ◽  
Critical Pair ◽  
Cyclic Groups ◽  
Free Subset ◽  
Free Sets

A subset$A$of a finite abelian group$G$is called$(k,l)$-sum-free if the sum of$k$(not necessarily distinct) elements of$A$never equals the sum of$l$(not necessarily distinct) elements of $A$. We find an explicit formula for the maximum size of a$(k,l)$-sum-free subset in$G$for all$k$and$l$in the case when$G$is cyclic by proving that it suffices to consider$(k,l)$-sum-free intervals in subgroups of $G$. This simplifies and extends earlier results by Hamidoune and Plagne [‘A new critical pair theorem applied to sum-free sets in abelian groups’,Comment. Math. Helv. 79(1) (2004), 183–207] and Bajnok [‘On the maximum size of a$(k,l)$-sum-free subset of an abelian group’,Int. J. Number Theory 5(6) (2009), 953–971].

scholarly journals On finite loops whose inner mapping groups are Abelian II

2005 ◽  
Vol 71 (3) ◽  
pp. 487-492
Author(s):  
Markku Niemenmaa
Keyword(s):  
Abelian Group ◽  
Direct Product ◽  
Prime Power ◽  
Abelian Groups ◽  
Finite Abelian Group ◽  
Prime Power Order ◽  
Finite Abelian Groups ◽  
Cyclic Groups ◽  
Inner Mapping Group ◽  
Mapping Group

If the inner mapping group of a loop is a finite Abelian group, then the loop is centrally nilpotent. We first investigate the structure of those finite Abelian groups which are not isomorphic to inner mapping groups of loops and after this we show that if the inner mapping group of a loop is isomorphic to the direct product of two cyclic groups of the same odd prime power order pn, then our loop is centrally nilpotent of class at most n + 1.

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