Incidence Matrices and Linear Graphs

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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An inequality for incidence matrices

Proceedings of the American Mathematical Society ◽

10.1090/s0002-9939-1959-0104592-8 ◽

1959 ◽

Vol 10 (2) ◽

pp. 216-216 ◽

Cited By ~ 6

Author(s):

J. R. Isbell

Keyword(s):

Incidence Matrices

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Logarithmic Axis Graphs Distort Lay Judgment

10.31234/osf.io/cwt56 ◽

2020 ◽

Cited By ~ 1

Author(s):

William Ryan ◽

Ellen Riemke Katrien Evers

Keyword(s):

Logarithmic Scale ◽

Linear Scale ◽

Policy Interventions ◽

Linear Graphs

COVID-19 data is often presented using graphs with either a linear or logarithmic scale. Given the importance of this information, understanding how choice of scale changes interpretations is critical. To test this, we presented laypeople with the same data plotted using differing scales. We found that graphs with a logarithmic, as opposed to linear, scale resulted in laypeople making less accurate predictions of growth, viewing COVID-19 as less dangerous, and expressing both less support for policy interventions and less intention to take personal actions to combat COVID-19. Education reduces, but does not eliminate these effects. These results suggest that public communications should use logarithmic graphs only when necessary, and such graphs should be presented alongside education and linear graphs of the same data whenever possible.

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Isotopy invariants of topological spaces

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

10.1098/rspa.1960.0071 ◽

1960 ◽

Vol 255 (1282) ◽

pp. 331-366 ◽

Cited By ~ 10

Keyword(s):

Homotopy Type ◽

Algebraic Topology ◽

Complete System ◽

Topological Spaces ◽

Recent Developments ◽

The Difference ◽

Linear Graphs ◽

Homotopy Invariants ◽

General Method

A few decades ago, topologists had already emphasized the difference between homotopy and isotopy. However, recent developments in algebraic topology are almost exclusively on the side of homotopy. Since a complete system of homotopy invariants has been obtained by Postnikov, it seems that hereafter we should pay more attention to isotopy invariants and new efforts should be made to attack the classical problems. The purpose of this paper is to introduce and study new algebraic isotopy invariants of spaces. A general method of constructing these invariants is given by means of a class of functors called isotopy functors. Special isotopy functors are constructed in this paper, namely, the m th residual functor R m and the m th. enveloping functor E m . Applications of these isotopy invariants to linear graphs are given in the last two sections. It turns out that these invariants can distinguish various spaces belonging to the same homotopy type.

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