scholarly journals Some properties and applications of Hermitian varieties in a finite projective space PG(N, q2) in the construction of strongly regular graphs (two-class association schemes) and block designs

1971 ◽  
Vol 11 (3) ◽  
pp. 268-283 ◽  
Author(s):  
I.M Chakravarti
Keyword(s):  
Projective Space ◽  
Association Schemes ◽  
Finite Projective Space ◽  
Strongly Regular Graphs ◽  
Block Designs ◽  
Regular Graphs ◽  
Strongly Regular

scholarly journals Cyclotomic Trace Codes

Algorithms ◽  
2019 ◽  
Vol 12 (8) ◽  
pp. 168
Author(s):  
Dean Crnković ◽  
Andrea Švob ◽  
Vladimir D. Tonchev
Keyword(s):  
Quantum Codes ◽  
Strongly Regular Graphs ◽  
Block Designs ◽  
Regular Graphs ◽  
Combinatorial Structures ◽  
Trace Codes ◽  
Strongly Regular ◽  
Projective Codes

A generalization of Ding’s construction is proposed that employs as a defining set the collection of the sth powers ( s ≥ 2 ) of all nonzero elements in G F ( p m ) , where p ≥ 2 is prime. Some of the resulting codes are optimal or near-optimal and include projective codes over G F ( 4 ) that give rise to optimal or near optimal quantum codes. In addition, the codes yield interesting combinatorial structures, such as strongly regular graphs and block designs.

scholarly journals Switching of Edges in Strongly Regular Graphs I: A Family of Partial Difference Sets on 100 Vertices

10.37236/1710 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
L. K. Jørgensen ◽  
M. Klin
Keyword(s):  
Automorphism Groups ◽  
Difference Sets ◽  
Association Schemes ◽  
Strongly Regular Graphs ◽  
Regular Graphs ◽  
Partial Difference ◽  
Galois Correspondence ◽  
Partial Difference Sets ◽  
Strongly Regular ◽  
Open Question

We present 15 new partial difference sets over 4 non-abelian groups of order 100 and 2 new strongly regular graphs with intransitive automorphism groups. The strongly regular graphs and corresponding partial difference sets have the following parameters: (100,22,0,6), (100,36,14,12), (100,45,20,20), (100,44,18,20). The existence of strongly regular graphs with the latter set of parameters was an open question. Our method is based on combination of Galois correspondence between permutation groups and association schemes, classical Seidel's switching of edges and essential use of computer algebra packages. As a by-product, a few new amorphic association schemes with 3 classes on 100 points are discovered.

Strongly Regular Graphs Derived from Combinatorial Designs

1970 ◽  
Vol 22 (3) ◽  
pp. 597-614 ◽  
Author(s):  
J. M. Goethals ◽  
J. J. Seidel
Keyword(s):  
Strongly Regular Graph ◽  
Hadamard Matrices ◽  
Latin Squares ◽  
Discrete Mathematics ◽  
Strongly Regular Graphs ◽  
Block Designs ◽  
Regular Graphs ◽  
Geometric Configurations ◽  
Strongly Regular ◽  
Definition Of

Several concepts in discrete mathematics such as block designs, Latin squares, Hadamard matrices, tactical configurations, errorcorrecting codes, geometric configurations, finite groups, and graphs are by no means independent. Combinations of these notions may serve the development of any one of them, and sometimes reveal hidden interrelations. In the present paper a central role in this respect is played by the notion of strongly regular graph, the definition of which is recalled below.In § 2, a fibre-type construction for graphs is given which, applied to block designs withλ= 1 and Hadamard matrices, yields strongly regular graphs. The method, although still limited in its applications, may serve further developments. In § 3 we deal with block designs, first considered by Shrikhande[22],in which the number of points in the intersection of any pair of blocks attains only two values.

scholarly journals Uniform Mixing and Association Schemes

10.37236/4745 ◽  
2017 ◽  
Vol 24 (3) ◽  
Author(s):  
Chris Godsil ◽  
Natalie Mullin ◽  
Aidan Roy
Keyword(s):  
Continuous Time ◽  
Hadamard Matrices ◽  
Quantum Walks ◽  
Association Schemes ◽  
Strongly Regular Graphs ◽  
Regular Graphs ◽  
Paley Graph ◽  
Time Quantum ◽  
Strongly Regular ◽  
Distance Regular Graphs

We consider continuous-time quantum walks on distance-regular graphs. Using results about the existence of complex Hadamard matrices in association schemes, we determine which of these graphs have quantum walks that admit uniform mixing.First we apply a result due to Chan to show that the only strongly regular graphs that admit instantaneous uniform mixing are the Paley graph of order nine and certain graphs corresponding to regular symmetric Hadamard matrices with constant diagonal. Next we prove that if uniform mixing occurs on a bipartite graph $X$ with $n$ vertices, then $n$ is divisible by four. We also prove that if $X$ is bipartite and regular, then $n$ is the sum of two integer squares. Our work on bipartite graphs implies that uniform mixing does not occur on $C_{2m}$ for $m \geq 3$. Using a result of Haagerup, we show that uniform mixing does not occur on $C_p$ for any prime $p$ such that $p \geq 5$. In contrast to this result, we see that $\epsilon$-uniform mixing occurs on $C_p$ for all primes $p$.

scholarly journals Two-weight and three-weight linear codes constructed from Weil sums

2022 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Tonghui Zhang ◽  
Hong Lu ◽  
Shudi Yang
Keyword(s):  
Secret Sharing ◽  
Linear Codes ◽  
Association Schemes ◽  
Strongly Regular Graphs ◽  
Secret Sharing Schemes ◽  
Regular Graphs ◽  
Authentication Codes ◽  
Strongly Regular ◽  
Weil Sums ◽  
Defining Sets

<p style='text-indent:20px;'>Linear codes with few weights are widely used in strongly regular graphs, secret sharing schemes, association schemes and authentication codes. In this paper, we construct several two-weight and three-weight linear codes over finite fields by choosing suitable different defining sets. We also give some examples and some of the codes are optimal or almost optimal. Their applications to secret sharing schemes are also investigated.</p>

Sign in / Sign up

Export Citation Format

Share Document