Incidence matrices and inequalities for combinatorial designs

2001 ◽  
Vol 10 (1) ◽  
pp. 17-26 ◽  
Author(s):  
D. Raghavarao ◽  
S.S. Shrikhande ◽  
M.S. Shrikhande
Keyword(s):  
Combinatorial Designs ◽  
Incidence Matrices

Flexible construction of measurement matrices in compressed sensing based on extensions of incidence matrices of combinatorial designs

2022 ◽  
Vol 420 ◽  
pp. 126901
Author(s):  
Junying Liang ◽  
Haipeng Peng ◽  
Lixiang Li ◽  
Fenghua Tong ◽  
Yixian Yang
Keyword(s):  
Compressed Sensing ◽  
Combinatorial Designs ◽  
Incidence Matrices

scholarly journals Hadamard Matrices with Cocyclic Core

Mathematics ◽  
2021 ◽  
Vol 9 (8) ◽  
pp. 857
Author(s):  
Víctor Álvarez ◽  
José Andrés Armario ◽  
María Dolores Frau ◽  
Félix Gudiel ◽  
María Belén Güemes ◽  
...  
Keyword(s):  
Hadamard Matrices ◽  
Combinatorial Designs ◽  
Powerful Technique ◽  
Uniform Approach ◽  
Structured Approach

Since Horadam and de Launey introduced the cocyclic framework on combinatorial designs in the 1990s, it has revealed itself as a powerful technique for looking for (cocyclic) Hadamard matrices. Ten years later, the series of papers by Kotsireas, Koukouvinos and Seberry about Hadamard matrices with one or two circulant cores introduced a different structured approach to the Hadamard conjecture. This paper is built on both strengths, so that Hadamard matrices with cocyclic cores are introduced and studied. They are proved to strictly include usual Hadamard matrices with one and two circulant cores, and therefore provide a wiser uniform approach to a structured Hadamard conjecture.

Constructing Byzantine Quorum systems from combinatorial designs

1999 ◽  
Vol 71 (1) ◽  
pp. 35-42 ◽  
Author(s):  
Tatsuhiro Tsuchiya ◽  
Nobuhiko Ido ◽  
Tohru Kikuno
Keyword(s):  
Combinatorial Designs ◽  
Quorum Systems

Several classes of permutation trinomials from Niho exponents over finite fields of characteristic 3

2019 ◽  
Vol 18 (04) ◽  
pp. 1950069
Author(s):  
Qian Liu ◽  
Yujuan Sun
Keyword(s):  
Positive Integer ◽  
Finite Field ◽  
Coding Theory ◽  
Permutation Polynomials ◽  
Combinatorial Designs ◽  
Dickson Polynomial ◽  
Niho Exponents ◽  
Characteristic 3 ◽  
Permutation Trinomials ◽  
Algebraic Forms

Permutation polynomials have important applications in cryptography, coding theory, combinatorial designs, and other areas of mathematics and engineering. Finding new classes of permutation polynomials is therefore an interesting subject of study. Permutation trinomials attract people’s interest due to their simple algebraic forms and additional extraordinary properties. In this paper, based on a seventh-degree and a fifth-degree Dickson polynomial over the finite field [Formula: see text], two conjectures on permutation trinomials over [Formula: see text] presented recently by Li–Qu–Li–Fu are partially settled, where [Formula: see text] is a positive integer.

scholarly journals Combinatorial designs and related computational constructions

1996 ◽  
Vol 53 (3) ◽  
pp. 521-523
Author(s):  
Peter Adams
Keyword(s):  
Combinatorial Designs

Combinatorial Designs

2016 ◽  
pp. 558-593
Keyword(s):  
Combinatorial Designs

scholarly journals Certain counterexamples to the construction of combinatorial designs on infinite sets.

1971 ◽  
Vol 12 (4) ◽  
pp. 461-466 ◽  
Author(s):  
William J. Frascella
Keyword(s):  
Combinatorial Designs ◽  
Infinite Sets

scholarly journals Some Pairwise Balanced Designs

10.37236/1491 ◽  
1999 ◽  
Vol 7 (1) ◽  
Author(s):  
Malcolm Greig
Keyword(s):  
Block Design ◽  
Basis Sets ◽  
Combinatorial Designs ◽  
Pairwise Balanced Design ◽  
Balanced Design ◽  
Pairwise Balanced Designs ◽  
Balanced Designs ◽  
Block Sizes

A pairwise balanced design, $B(K;v)$, is a block design on $v$ points, with block sizes taken from $K$, and with every pair of points occurring in a unique block; for a fixed $K$, $B(K)$ is the set of all $v$ for which a $B(K;v)$ exists. A set, $S$, is a PBD-basis for the set, $T$, if $T=B(S)$. Let $N_{a(m)}=\{n:n\equiv a\bmod m\}$, and $N_{\geq m}=\{n:n\geq m\}$; with $Q$ the corresponding restriction of $N$ to prime powers. This paper addresses the existence of three PBD-basis sets. 1. It is shown that $Q_{1(8)}$ is a basis for $N_{1(8)}\setminus E$, where $E$ is a set of 5 definite and 117 possible exceptions. 2. We construct a 78 element basis for $N_{1(8)}$ with, at most, 64 inessential elements. 3. Bennett and Zhu have shown that $Q_{\geq8}$ is a basis for $N_{\geq8}\setminus E'$, where $E'$ is a set of 43 definite and 606 possible exceptions. Their result is improved to 48 definite and 470 possible exceptions. (Constructions for 35 of these possible exceptions are known.) Finally, we provide brief details of some improvements and corrections to the generating/exception sets published in The CRC Handbook of Combinatorial Designs.

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.

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