On the Number of Slopes of the Graph of a Function Defined on a Finite Field

Blokhuis, Aart; Ball, Simeon; Brouwer, Andries E.; Storme, Leo; Szoőnyi, Tamás

doi:10.1006/jcta.1998.2915

Cited by 86 publications

(101 citation statements)

References 3 publications

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“…The problem was first examined Blockhuis, Ball, Brouwer, Storme and Szonyi [6], then by Ball [7] [8] who has done an extensive investigation. However, there has not been any attempt on computing the directions themselves, so far.…”

Section: And a Point A Rmentioning

confidence: 99%

Derivatives over Certain Finite Rings

Mohamed¹

2017

OALib

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We introduce a derivative of a relation over the ring of integers modulo an odd number which is based on the very fundamental concepts which helped in the evolution of derivative of a function over the real number field, namely slope. Then, for a prime field ( ) GF p , we use the derivatives to construct an algorithm that find all the directions, in the sense of Redei [1], of graphs of certain exponential relations over R.

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Section: And a Point A Rmentioning

confidence: 99%

Derivatives over Certain Finite Rings

Mohamed¹

2017

OALib

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“…(iv) (Blokhuis, Ball, Brouwer, Storme, Szőnyi [6], Ball [2]) if p > 2 and there exists a line intersecting B in |B ∩ | = |B| − q points (so a blocking set of Rédei type);…”

Section: Theorem 12 (Ball [1])mentioning

confidence: 99%

A characterization of multiple (n – k)-blocking sets in projective spaces of square order

Ferret

Storme

Sziklai

et al. 2012

Advances in Geometry

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“…Under these hypotheses for q, the non-trivial minimal blocking sets of PG(2, q 3 ) of the smallest and second smallest size have been completely classified in [7,61,63,70].…”

Section: Maximal Partial Spreadsmentioning

confidence: 99%

The Use of Blocking Sets in Galois Geometries and in Related Research Areas

Pepe

Storme

2011

Buildings, Finite Geometries and Groups

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Blocking sets play a central role in Galois geometries. Besides their intrinsic geometrical importance, the importance of blocking sets also arises from the use of blocking sets for the solution of many other geometrical problems, and problems in related research areas. This article focusses on these applications to motivate researchers to investigate blocking sets, and to motivate researchers to investigate the problems that can be solved by using blocking sets. By showing the many applications on blocking sets, we also wish to prove that researchers who improve results on blocking sets in fact open the door to improvements on the solution of many other problems. Definitions and introductory resultsA set B of points of PG(n, q) is called a k-blocking set if every (n − k)-dimensional subspace of PG(n, q) has a non-empty intersection with B, and a set B of points of PG(n, q) is called a t-fold k-blocking set if every (n − k)-dimensional subspace contains at least t points of B. In PG(2, q), the 1-blocking sets are simply called blocking sets. Two subspaces Σ 1 and Σ 2 of PG(n, q) of dimension k and n − k always have a non-empty intersection, hence a set B containing a k-dimensional subspace is called a trivial k-blocking set. Moreover, a k-blocking set B is called minimal if it is minimal with respect to the containment relation, i.e. B \ {P } is not a k-blocking set for every P ∈ B, and it is called small if |B| < 3(q k +1) 2 . The blocking sets of PG(2, q) have been extensively studied in the last years and there are plenty of results about characterizations (of small ones) and about the bounds on the size. A trivial blocking set of PG(2, q) is a set of points containing a line. The first examples of non-trivial blocking sets of small size of PG(2, q) have been given in the following: Theorem 1. In PG(2, q), q odd, there exists a projective triangle of side q+3 2 that is a minimal blocking set of size[14]. In PG(2, q), q even, there exists a projective triad of side q+2 2 that is a minimal blocking set of size 3q+2 2[40].This motivates the choice of the boundfor the size of a small blocking set in PG(2, q) and then generalized asfor the k-blocking sets of PG(n, q).

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On the Number of Slopes of the Graph of a Function Defined on a Finite Field

Abstract: Given a set U of size q in an affine plane of order q, we determine the possibilities for the number of directions of secants of U, and in many cases characterize the sets U with given number of secant directions. Academic Press

Cited by 86 publications

References 3 publications

Derivatives over Certain Finite Rings

Derivatives over Certain Finite Rings

A characterization of multiple (n – k)-blocking sets in projective spaces of square order

The Use of Blocking Sets in Galois Geometries and in Related Research Areas

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