A canonical form for incidence matrices of finite projective planes and their associated latin squares and planar ternary rings

1983 ◽  
pp. 111-120
Author(s):  
Stephen Bourn
Keyword(s):  
Canonical Form ◽  
Projective Planes ◽  
Latin Squares ◽  
Incidence Matrices ◽  
Finite Projective Planes

The Non-Existence of Finite Projective Planes of Order 10

1989 ◽  
Vol 41 (6) ◽  
pp. 1117-1123 ◽  
Author(s):  
C. W. H. Lam ◽  
L. Thiel ◽  
S. Swiercz
Keyword(s):  
Projective Plane ◽  
Prime Power ◽  
Projective Planes ◽  
Latin Squares ◽  
Orthogonal Latin Squares ◽  
Finite Projective Planes

A finite projective plane of order n, with n > 0, is a collection of n2+ n + 1 lines and n2+ n + 1 points such that1. every line contains n + 1 points,2. every point is on n + 1 lines,3. any two distinct lines intersect at exactly one point, and4. any two distinct points lie on exactly one line.It is known that a plane of order n exists if n is a prime power. The first value of n which is not a prime power is 6. Tarry [18] proved in 1900 that a pair of orthogonal latin squares of order 6 does not exist, which by Bose's 1938 result [3] implies that a projective plane of order 6 does not exist.

Combinatorial Problems

1950 ◽  
Vol 2 ◽  
pp. 93-99 ◽  
Author(s):  
S. Chowla ◽  
H. J. Ryser
Keyword(s):  
Combinatorial Problem ◽  
Difference Sets ◽  
Projective Planes ◽  
Combinatorial Problems ◽  
Block Designs ◽  
Incomplete Block ◽  
Incidence Matrices ◽  
Finite Projective Planes ◽  
Balanced Incomplete Block Designs ◽  
Balanced Incomplete Block

Let it be required to arrange v elements into v sets such that every set contains exactly k distinct elements and such that every pair of sets has exactly elements in common . This combinatorial problem is studied in conjunction with several similar problems, and these problems are proved impossible for an infinitude of v and k. An incidence matrix is associated with each of the combinatorial problems, and the problems are then studied almost entirely in terms of their incidence matrices. The techniques used are similar to those developed by Bruck and Ryser for finite projective planes [3]. The results obtained are of significance in the study of Hadamard matrices [6;8], finite projective planes [9], symmetrical balanced incomplete block designs [2; 5], and difference sets [7].

scholarly journals Embedding (r, 1)-designs in finite projective planes

1977 ◽  
Vol 19 (1) ◽  
pp. 67-76 ◽  
Author(s):  
D. McCarthy ◽  
S.A. Vanstone
Keyword(s):  
Projective Planes ◽  
Finite Projective Planes

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.

scholarly journals Analysis of Some Properties from Construction of Finite Projective Planes of Order 2 and 3

CAUCHY ◽  
2016 ◽  
Vol 4 (3) ◽  
pp. 131
Author(s):  
Vira Hari Krisnawati ◽  
Corina Karim
Keyword(s):  
Automorphism Group ◽  
Projective Plane ◽  
Symmetric Group ◽  
Block Design ◽  
Finite Projective Plane ◽  
Projective Planes ◽  
Steiner System ◽  
Incidence Relation ◽  
Finite Projective Planes ◽  
Set Of Points

<p class="abstract"><span lang="IN">In combinatorial mathematics, a Steiner system is a type of block design. Specifically, a Steiner system <em>S</em>(<em>t</em>, <em>k</em>, <em>v</em>) is a set of <em>v</em> points and <em>k</em> blocks which satisfy that every <em>t</em>-subset of <em>v</em>-set of points appear in the unique block. It is well-known that a finite projective plane is one examples of Steiner system with <em>t</em> = 2, which consists of a set of points and lines together with an incidence relation between them and order 2 is the smallest order.</span></p><p class="abstract"><span lang="IN">In this paper, we observe some properties from construction of finite projective planes of order 2 and 3. Also, we analyse the intersection between two projective planes by using some characteristics of the construction and orbit of projective planes over some representative cosets from automorphism group in the appropriate symmetric group.</span></p>

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