scholarly journals Increasing the Minimum Distance of Codes by Twisting

10.37236/5852 ◽  
2018 ◽  
Vol 25 (3) ◽  
Author(s):  
Marzieh Akbari ◽  
Neil I. Gillespie ◽  
Cheryl E. Praeger
Keyword(s):  
Finite Groups ◽  
Minimum Distance ◽  
Symplectic Groups ◽  
Permutation Codes ◽  
Graph Automorphism ◽  
Permutation Code ◽  
Permutation Arrays ◽  
The Family ◽  
Even Order ◽  
Minimum Distances

Twisted permutation codes, introduced recently by the second and third authors, belong to the family of frequency permutation arrays. Like some other codes in this family, such as the repetition permutation codes, they are obtained by a repetition construction applied to a smaller code (but with a "twist" allowed). The minimum distance of a twisted permutation code is known to be at least the minimum distance of a corresponding repetition permutation code, but in some instances can be larger. We construct two new infinite families of twisted permutation codes with minimum distances strictly greater than those for the corresponding repetition permutation codes. These constructions are based on two infinite families of finite groups and their representations. The first is a family of $p$-groups, for an odd prime $p$, while the second family consists of the $4$-dimensional symplectic groups over a finite field of even order. In the latter construction, properties of the graph automorphism of these symplectic groups play an important role.

scholarly journals A Bound on Permutation Codes

10.37236/2929 ◽  
2013 ◽  
Vol 20 (3) ◽  
Author(s):  
Jürgen Bierbrauer ◽  
Klaus Metsch
Keyword(s):  
Projective Plane ◽  
Symmetric Group ◽  
Minimum Distance ◽  
Main Part ◽  
Latin Squares ◽  
Permutation Codes ◽  
Permutation Code ◽  
Mutually Orthogonal Latin Squares ◽  
Hamming Metric ◽  
Completion Theorem

Consider the symmetric group $S_n$ with the Hamming metric. A  permutation code on $n$ symbols is a subset $C\subseteq S_n.$ If $C$ has minimum distance $\geq n-1,$ then $\vert C\vert\leq n^2-n.$ Equality can be reached if and only if a projective plane of order $n$ exists. Call $C$ embeddable if it is contained in a permutation code of minimum distance $n-1$ and cardinality $n^2-n.$ Let $\delta =\delta (C)=n^2-n-\vert C\vert$ be the deficiency of the permutation code $C\subseteq S_n$ of minimum distance $\geq n-1.$We prove that $C$ is embeddable if either $\delta\leq 2$ or if $(\delta^2-1)(\delta +1)^2<27(n+2)/16.$ The main part of the proof is an adaptation of the method used to obtain the famous Bruck completion theorem for mutually orthogonal latin squares.

scholarly journals Nonabelian Sylow subgroups of finite groups of even order

1998 ◽  
Vol 4 (12) ◽  
pp. 88-90
Author(s):  
Naoki Chigira ◽  
Nobuo Iiyori ◽  
Hiroyoshi Yamaki
Keyword(s):  
Finite Groups ◽  
Sylow Subgroups ◽  
Even Order

scholarly journals A new efficient way based on special stabilizer multiplier permutations to attack the hardness of the minimum weight search problem for large BCH codes

2019 ◽  
Vol 9 (2) ◽  
pp. 1232
Author(s):  
Issam Abderrahman Joundan ◽  
Said Nouh ◽  
Mohamed Azouazi ◽  
Abdelwahed Namir
Keyword(s):  
Coding Theory ◽  
Minimum Distance ◽  
Minimum Weight ◽  
Error Correcting Codes ◽  
Bch Codes ◽  
Search Problem ◽  
Public Key Cryptosystems ◽  
True Value ◽  
Efficient Scheme ◽  
Minimum Distances

<span>BCH codes represent an important class of cyclic error-correcting codes; their minimum distances are known only for some cases and remains an open NP-Hard problem in coding theory especially for large lengths. This paper presents an efficient scheme ZSSMP (Zimmermann Special Stabilizer Multiplier Permutation) to find the true value of the minimum distance for many large BCH codes. The proposed method consists in searching a codeword having the minimum weight by Zimmermann algorithm in the sub codes fixed by special stabilizer multiplier permutations. These few sub codes had very small dimensions compared to the dimension of the considered code itself and therefore the search of a codeword of global minimum weight is simplified in terms of run time complexity.  ZSSMP is validated on all BCH codes of length 255 for which it gives the exact value of the minimum distance. For BCH codes of length 511, the proposed technique passes considerably the famous known powerful scheme of Canteaut and Chabaud used to attack the public-key cryptosystems based on codes. ZSSMP is very rapid and allows catching the smallest weight codewords in few seconds. By exploiting the efficiency and the quickness of ZSSMP, the true minimum distances and consequently the error correcting capability of all the set of 165 BCH codes of length up to 1023 are determined except the two cases of the BCH(511,148) and BCH(511,259) codes. The comparison of ZSSMP with other powerful methods proves its quality for attacking the hardness of minimum weight search problem at least for the codes studied in this paper.</span>

scholarly journals Maximal Arcs, Codes, and New Links Between Projective Planes of Order 16

10.37236/9008 ◽  
2020 ◽  
Vol 27 (1) ◽  
Author(s):  
Mustafa Gezek ◽  
Rudi Mathon ◽  
Vladimir D. Tonchev
Keyword(s):  
Minimum Distance ◽  
Linear Codes ◽  
Projective Planes ◽  
Upper And Lower Bounds ◽  
Dual Codes ◽  
Binary Linear Codes ◽  
Incidence Matrices ◽  
Maximal Arc ◽  
Maximal Arcs ◽  
Even Order

In this paper we consider binary linear codes spanned by incidence matrices of Steiner 2-designs associated with maximal arcs in projective planes of even order, and their dual codes. Upper and lower bounds on the 2-rank of the incidence matrices are derived. A lower bound on the minimum distance of the dual codes is proved, and it is shown that the bound is achieved if and only if the related maximal arc contains a hyperoval of the plane. The  binary linear codes of length 52 spanned by the incidence matrices of 2-$(52,4,1)$ designs associated with previously known and some newly found maximal arcs of degree 4 in projective planes of order 16 are analyzed and classified up to equivalence. The classification shows that some designs associated with maximal arcs in nonisomorphic planes generate equivalent codes. This phenomenon establishes new links between several of the known planes. A conjecture concerning the codes of maximal arcs in $PG(2,2^m)$ is formulated.

On the Frobenius spectrum of a finite group

2017 ◽  
Vol 16 (03) ◽  
pp. 1750051 ◽  
Author(s):  
Jiangtao Shi ◽  
Wei Meng ◽  
Cui Zhang
Keyword(s):  
Finite Group ◽  
Positive Integer ◽  
Finite Groups ◽  
Prime Divisor ◽  
Complete Classification ◽  
Composition Factor ◽  
Frobenius Theorem ◽  
Even Order ◽  
A Finite Group

Let [Formula: see text] be a finite group and [Formula: see text] any divisor of [Formula: see text], the order of [Formula: see text]. Let [Formula: see text], Frobenius’ theorem states that [Formula: see text] for some positive integer [Formula: see text]. We call [Formula: see text] a Frobenius quotient of [Formula: see text] for [Formula: see text]. Let [Formula: see text] be the set of all Frobenius quotients of [Formula: see text], we call [Formula: see text] the Frobenius spectrum of [Formula: see text]. In this paper, we give a complete classification of finite groups [Formula: see text] with [Formula: see text] for [Formula: see text] being the smallest prime divisor of [Formula: see text]. Moreover, let [Formula: see text] be a finite group of even order, [Formula: see text] the set of all Frobenius quotients of [Formula: see text] for even divisors of [Formula: see text] and [Formula: see text] the maximum Frobenius quotient in [Formula: see text], we prove that [Formula: see text] is always solvable if [Formula: see text] or [Formula: see text] and [Formula: see text] is not a composition factor of [Formula: see text].

Class-Preserving Coleman Automorphisms of Finite Groups with Prescribed Centralizers

2017 ◽  
Vol 24 (02) ◽  
pp. 351-360
Author(s):  
Zhengxing Li ◽  
Hongwei Gao
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Inner Automorphism ◽  
Coleman Automorphism ◽  
Even Order ◽  
A Finite Group

Let G be a finite group. It is proved that any class-preserving Coleman automorphism of G is an inner automorphism whenever G belongs to one of the following two classes of groups: (1) CN-groups, i.e., groups in which the centralizer of any element is nilpotent; (2) CIT-groups, i.e., groups of even order in which the centralizer of any involution is a 2-group. In particular, the normalizer conjecture holds for both CN-groups and CIT-groups. Additionally, some other results are also obtained.

scholarly journals Determination of the critical distance in the procedure of explosive welding

2020 ◽  
Vol 68 (4) ◽  
pp. 823-844
Author(s):  
Miloš Lazarević ◽  
Bogdan Nedić ◽  
Jovica Bogdanov ◽  
Stefan Đurić
Keyword(s):  
Hearing Loss ◽  
Explosive Welding ◽  
Minimum Distance ◽  
Critical Distance ◽  
Maximum Pressure ◽  
Limit Pressure ◽  
Minimum Distances ◽  
Wave Methods ◽  
Welding Procedure

Introduction/purpose: When performing the explosive welding procedure, for the safety of workers, it is necessary to take into account the minimum distance between the workers and the place of explosion at the time of explosion. Negligence can cause temporary hearing loss, rupture of the eardrum and in some cases even the death of workers. The aim of this paper is to determine the critical distance based on the mass of explosive charge required for explosive welding, provided that the limit pressure is 6.9 kPa in the case of temporary hearing loss and 35 kPa in the case of eardrum rupture. This paper does not take into account other effects of the explosion than those caused by the shock wave. Methods: Depending on the type of explosion, the equivalent explosive mass was calculated. Based on the equivalent explosive mass and the limit pressure, the minimum distances were calculated using the Sadovsky and Kingery-Bulmash equations. Results: The corresponding tables show the results of the calculation of the critical distance of workers from the place of the explosion when there may be temporary hearing loss or rupture of the eardrum. The calculated value of the critical explosion distance by the Kingery-Bulmash method, under the condition of the maximum pressure for temporary hearing loss, is 5.62% lower than the distance value obtained by the Sadovsky method while the value of the critical explosion distance calculated by the Kingery-Bulmash method, under the condition of the maximum pressure for eardrum rupture, is 7.83% lower than the value obtained by the Sadovsky method. Conclusion: The results of the calculation showed that the critical distance from the explosion can be successfully calculated and that the obtained values have small differences depending on the applied calculation method.

scholarly journals Assessment of the capabilities of orbital optical devices for obtaining information about space objects

2021 ◽  
Vol 20 (2) ◽  
pp. 270-301
Author(s):  
Valeriy Prorok ◽  
Anatoliy Karytko ◽  
Alexander Goryanskiy ◽  
Ekaterina Emelyanova
Keyword(s):  
Minimum Distance ◽  
Fuzzy Inference ◽  
Optical Devices ◽  
Time Interval ◽  
Optimal Conditions ◽  
Relative Speed ◽  
Space Objects ◽  
Research Problems ◽  
Minimum Distances ◽  
Minimum Points

The purpose of the study is to select the optimal conditions for collecting non-coordinate information about a spacecraft by a space optical-electronic means at the time objects pass the vicinity of the points of the minimum distance between their orbits. The quantitative indicator is proposed that characterize the measure of the possibility of obtaining non-coordinate information about space objects with the required level of quality. The arguments of the function characterizing the indicator are the distance between spacecraft; their relative speed; phase angle of illumination of a spacecraft by the Sun in relation to the optical-electronic means; the length of the time interval during which both objects are in the vicinity of the point of a minimum distance between their orbits. The value of the indicator is computed by solving three particular research problems. The first task is to search for neighborhoods that include the minimum distances between the orbits of the controlled spacecraft and optical-electronic means. To solve it, a fast algorithm for calculating the minimum distance between orbits used. Additionally, the drift of the found neighborhoods is taken into account on the time interval up to 60 hours. The second task is to estimate the characteristics of motion and the conditions of optical visibility of a controlled spacecraft in the vicinity of the minimum points of the distance between the orbits of spacecraft. The solution to this problem is carried out by using the SGP4 library of space objects motion forecast. The third task is justification and calculation of an index characterizing the measure of the possibility of obtaining an optical image of a spacecraft for given conditions of optical visibility. To solve the problem, the developed system of fuzzy inference rules and the Mamdani algorithm is used. The presented method is implemented as a program. In the course of a computational experiment, an assessment was made of the possibility of obtaining non-coordinate information on low-orbit and geostationary space objects. The proposed indicator provides an increase in the efficiency of the procedure for collecting non-coordinate information about space objects by choosing the most informative alternatives for monitoring space objects from the available set of possible observations at a given planning interval for collecting information about space objects.

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