scholarly journals Enumeration of Self-Dual Codes of Length 6 over ℤp

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 882
Author(s):  
Whan-Hyuk Choi
Keyword(s):  
Automorphism Group ◽  
Finite Field ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Mds Codes ◽  
Dual Codes ◽  
Necessary And Sufficient

The purpose of this paper is to classify and enumerate self-dual codes of length 6 over finite field Z p . First, we classify these codes into three cases: decomposable, indecomposable non-MDS and MDS codes. Then, we complete the classification of non-MDS self-dual codes of length 6 over Z p for all primes p in terms of their automorphism group. We obtain all inequivalent classes and find the necessary and sufficient conditions for the existence of each class. Finally, we obtain the number of MDS self-dual codes of length 6.

scholarly journals Hermitian Self-Orthogonal Constacyclic Codes over Finite Fields

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Amita Sahni ◽  
Poonam Trama Sehgal
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Mds Codes ◽  
Constacyclic Codes ◽  
Necessary And Sufficient ◽  
Defining Sets

Necessary and sufficient conditions for the existence of Hermitian self-orthogonal constacyclic codes of length n over a finite field Fq2, n coprime to q, are found. The defining sets and corresponding generator polynomials of these codes are also characterised. A formula for the number of Hermitian self-orthogonal constacyclic codes of length n over a finite field Fq2 is obtained. Conditions for the existence of numerous MDS Hermitian self-orthogonal constacyclic codes are obtained. The defining set and the number of such MDS codes are also found.

scholarly journals The Action of the Weyl Group on the $$E_8$$ Root System

2021 ◽  
Author(s):  
Rosa Winter ◽  
Ronald van Luijk
Keyword(s):  
Automorphism Group ◽  
Weyl Group ◽  
Root System ◽  
Sufficient Conditions ◽  
Maximal Clique ◽  
Necessary And Sufficient Conditions ◽  
Inner Product ◽  
Colored Graphs ◽  
Necessary And Sufficient ◽  
Root Polytope

AbstractLet $$\varGamma $$ Γ be the graph on the roots of the $$E_8$$ E 8 root system, where any two distinct vertices e and f are connected by an edge with color equal to the inner product of e and f. For any set c of colors, let $$\varGamma _c$$ Γ c be the subgraph of $$\varGamma $$ Γ consisting of all the 240 vertices, and all the edges whose color lies in c. We consider cliques, i.e., complete subgraphs, of $$\varGamma $$ Γ that are either monochromatic, or of size at most 3, or a maximal clique in $$\varGamma _c$$ Γ c for some color set c, or whose vertices are the vertices of a face of the $$E_8$$ E 8 root polytope. We prove that, apart from two exceptions, two such cliques are conjugate under the automorphism group of $$\varGamma $$ Γ if and only if they are isomorphic as colored graphs. Moreover, for an isomorphism f from one such clique K to another, we give necessary and sufficient conditions for f to extend to an automorphism of $$\varGamma $$ Γ , in terms of the restrictions of f to certain special subgraphs of K of size at most 7.

The λ-classification of continuous-time birth-and-death processes

2003 ◽  
Vol 35 (04) ◽  
pp. 1111-1130 ◽  
Author(s):  
Andrew G. Hart ◽  
Servet Martínez ◽  
Jaime San Martín
Keyword(s):  
Continuous Time ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Birth And Death Processes ◽  
Positive Recurrent ◽  
Necessary And Sufficient

We study the λ-classification of absorbing birth-and-death processes, giving necessary and sufficient conditions for such processes to be λ-transient, λ-null recurrent and λ-positive recurrent.

Asymptotic stability of heteroclinic cycles in systems with symmetry. II

2004 ◽  
Vol 134 (6) ◽  
pp. 1177-1197 ◽  
Author(s):  
Martin Krupa ◽  
Ian Melbourne
Keyword(s):  
Asymptotic Stability ◽  
Transition Matrix ◽  
Sufficient Conditions ◽  
Systematic Investigation ◽  
Necessary And Sufficient Conditions ◽  
Sufficient Condition ◽  
Heteroclinic Cycles ◽  
Higher Dimensional ◽  
Necessary And Sufficient

Systems possessing symmetries often admit robust heteroclinic cycles that persist under perturbations that respect the symmetry. In previous work, we began a systematic investigation into the asymptotic stability of such cycles. In particular, we found a sufficient condition for asymptotic stability, and we gave algebraic criteria for deciding when this condition is also necessary. These criteria are satisfied for cycles in R3.Field and Swift, and Hofbauer, considered examples in R4 for which our sufficient condition for stability is not optimal. They obtained necessary and sufficient conditions for asymptotic stability using a transition-matrix technique.In this paper, we combine our previous methods with the transition-matrix technique and obtain necessary and sufficient conditions for asymptotic stability for a larger class of heteroclinic cycles. In particular, we obtain a complete theory for ‘simple’ heteroclinic cycles in R4 (thereby proving and extending results for homoclinic cycles that were stated without proof by Chossat, Krupa, Melbourne and Scheel). A partial classification of simple heteroclinic cycles in R4 is also given. Finally, our stability results generalize naturally to higher dimensions and many of the higher-dimensional examples in the literature are covered by this theory.

Groups with prescribed automorphism group: A clarification

1984 ◽  
Vol 27 (1) ◽  
pp. 59-60
Author(s):  
Derek J. S. Robinson
Keyword(s):  
Automorphism Group ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Group A ◽  
Necessary And Sufficient ◽  
Central Automorphisms ◽  
Semisimple Subgroup

In Theorems 1 and 2 of [] necessary and sufficient conditions were given for a group G to have a finite automorphism group Aut G and a semisimple subgroup of central automorphisms AutcG. Recently it occurred to us, as a result of conversations with Ursula Webb, that these conditions could be stated in a much simpler and clearer form. Our purpose here is to record this reformulation. For an explanation ofterminology and notation we refer the reader to [1].

scholarly journals The Q0-matrix completion problem

2020 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Kalyan Sinha
Keyword(s):  
Matrix Completion ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Completion Problem ◽  
Matrix Completion Problem ◽  
Necessary And Sufficient ◽  
The Relationship

A matrix is a Q0-matrix if for every k∈{1,2,…,n}, the sum of all k×k principal minors is nonnegative. In this paper, we study some necessary and sufficient conditions for a digraph to have Q0-completion. Later on we discuss the relationship between Q and Q0-matrix completion problem. Finally, a classification of the digraphs of order up to four is done based on Q0-completion.

scholarly journals Asynchronous Computability Theorem in Arbitrary Solo Models

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 757
Author(s):  
Yunguang Yue ◽  
Fengchun Lei ◽  
Xingwu Liu ◽  
Jie Wu
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Computational Power ◽  
Nest Structure ◽  
Combinatorial Topology ◽  
Necessary And Sufficient ◽  
As If

In this paper, we establish the asynchronous computability theorem in d-solo system by borrowing concepts from combinatorial topology, in which we state a necessary and sufficient conditions for a task to be wait-free computable in that system. Intuitively, a d-solo system allows as many d processes to access it as if each were running solo, namely, without detecting communication from any peer. As an application, we completely characterize the solvability of the input-less tasks in such systems. This characterization also leads to a hardness classification of these tasks according to whether their output complexes hold a d-nest structure. As a byproduct, we find an alternative way to distinguish the computational power of d-solo objects for different d.

A classification of vertex-transitive Cayley digraphs of strong semilattices of completely simple semigroups

2012 ◽  
Vol 62 (5) ◽  
Author(s):  
Shou-feng Wang ◽  
Di Zhang
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Completely Simple Semigroups ◽  
Vertex Transitive ◽  
Necessary And Sufficient ◽  
Completely Simple

AbstractWith the help of a property of completely simple semigroups proved in this paper we give necessary and sufficient conditions for vertex-transitivity of Cayley digraphs of strong semilattices of completely simple semigroups.

scholarly journals On the Classification of Bol-Moufang Type of Some Varieties of Quasi Neutrosophic Triplet Loop (Fenyves BCI-Algebras)

Symmetry ◽  
2018 ◽  
Vol 10 (10) ◽  
pp. 427 ◽  
Author(s):  
Tèmítọ́pẹ́ Jaíyéọlá ◽  
Emmanuel Ilojide ◽  
Memudu Olatinwo ◽  
Florentin Smarandache
Keyword(s):  
Research Finding ◽  
Research Work ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Algebraic Structures ◽  
Associative Algebras ◽  
Necessary And Sufficient

In this paper, Bol-Moufang types of a particular quasi neutrosophic triplet loop (BCI-algebra), chritened Fenyves BCI-algebras are introduced and studied. 60 Fenyves BCI-algebras are introduced and classified. Amongst these 60 classes of algebras, 46 are found to be associative and 14 are found to be non-associative. The 46 associative algebras are shown to be Boolean groups. Moreover, necessary and sufficient conditions for 13 non-associative algebras to be associative are also obtained: p-semisimplicity is found to be necessary and sufficient for a F 3 , F 5 , F 42 and F 55 algebras to be associative while quasi-associativity is found to be necessary and sufficient for F 19 , F 52 , F 56 and F 59 algebras to be associative. Two pairs of the 14 non-associative algebras are found to be equivalent to associativity ( F 52 and F 55 , and F 55 and F 59 ). Every BCI-algebra is naturally an F 54 BCI-algebra. The work is concluded with recommendations based on comparison between the behaviour of identities of Bol-Moufang (Fenyves’ identities) in quasigroups and loops and their behaviour in BCI-algebra. It is concluded that results of this work are an initiation into the study of the classification of finite Fenyves’ quasi neutrosophic triplet loops (FQNTLs) just like various types of finite loops have been classified. This research work has opened a new area of research finding in BCI-algebras, vis-a-vis the emergence of 540 varieties of Bol-Moufang type quasi neutrosophic triplet loops. A ‘Cycle of Algebraic Structures’ which portrays this fact is provided.

scholarly journals SOME FAMILIES OF PERMUTATION POLYNOMIALS OVER FINITE FIELDS

2008 ◽  
Vol 04 (05) ◽  
pp. 851-857 ◽  
Author(s):  
MICHAEL E. ZIEVE
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Permutation Polynomials ◽  
Polynomials Over Finite Fields ◽  
Necessary And Sufficient

We give necessary and sufficient conditions for a polynomial of the form xr(1 + xv + x2v + ⋯ + xkv)t to permute the elements of the finite field 𝔽q. Our results yield especially simple criteria in case (q - 1)/ gcd (q - 1, v) is a small prime.

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