In this paper, we associate a weight function with a set of points satisfying the conditions of t... more In this paper, we associate a weight function with a set of points satisfying the conditions of the cylinder conjecture. Then we derive properties of this weight function using the Rédei polynomial of the point set. Using additional assumptions on the number of non-determined directions, together with an exhaustive computer search for weight functions satisfying particular properties, we prove a relaxed version of the cylinder conjecture for $$p \le 13$$p≤13. This result also slightly refines a result of Sziklai on point sets in $$\mathrm {AG}(3,p)$$AG(3,p).
Our word posets have finite words of bounded length as their elements, with the words composed fr... more Our word posets have finite words of bounded length as their elements, with the words composed from a finite alphabet. Their partial ordering follows from the inclusion of a word as a subsequence of another word. The elemental combinatorial properties of such posets are established. Their automorphism groups are determined (along with similar result for the word poset studied by Burosch, Frank and Röhl [4]) and a BLYM inequality is verified (via the normalized matching property).
Van Lint and MacWilliams (IEEE Trans. Inform. Theory IT 24 (1978) 730-737) conjectured that the o... more Van Lint and MacWilliams (IEEE Trans. Inform. Theory IT 24 (1978) 730-737) conjectured that the only q-subset X of GF(q 2), with the properties 0; 1 ∈ X and x − y is a square for all x; y ∈ X , is the set GF(q). Aart Blokhuis (Indag. Math. 46 (1984) 369-372) proved the conjecture for arbitrary odd q. In this paper we give a similar characterization of GF(q) in GF(q 2), proving the analogue of Blokhuis' theorem for dth powers (instead of squares), when d|(q + 1). We also prove an embedding-type result, stating that if |S| ¿ q − (1 − 1=d) √ q with the same properties as X above, then S ⊆ GF(q).
In [10], it was shown that small t-fold (n − k)-blocking sets in PG(n, q), q = p h , p prime, h ≥... more In [10], it was shown that small t-fold (n − k)-blocking sets in PG(n, q), q = p h , p prime, h ≥ 1, intersect every k-dimensional space in t (mod p) points. We characterize in this article all t-fold (n − k)-blocking sets in PG(n, q), q square, q ≥ 661, t < c p q 1/6 /2, |B| < tq n−k + 2tq n−k−1 √ q, intersecting every k-dimensional space in t (mod √ q) points. * The third author is grateful for the partial support of OTKA T049662, T067867 and Bolyai grants. In the theory of 1-fold planar blocking sets, 1 (mod p) results for small 1-fold planar blocking sets play an important role. Definition 1.3 A blocking set of PG(2, q) is called small when it has less than 3(q + 1)/2 points. If q = p h , p prime, h ≥ 1, the exponent e of the minimal blocking set B of PG(2, q) is the maximal integer e such that every line intersects B in 1 (mod p e) points. Theorem 1.4 Let B be a small minimal 1-fold blocking set in PG(2, q), q = p h , p prime, h ≥ 1. Then B intersects every line in 1 (mod p) points, so for the exponent e of B, we have 1 ≤ e ≤ h. (Szőnyi [18]) In fact, this exponent e is a divisor of h. (Sziklai [17]) This result was extended by Szőnyi and Weiner [19] to 1-fold (n − k)blocking sets in PG(n, q). Definition 1.5 A 1-fold (n−k)-blocking set of PG(n, q) is called small when it has less than 3(q n−k + 1)/2 points. If q = p h , p prime, h ≥ 1, the exponent e of the minimal 1-fold (n − k)blocking set B is the maximal integer e such that every hyperplane intersects B in 1 (mod p e) points. A most interesting question of the theory of blocking sets is to classify the small blocking sets. A natural construction (blocking the k-subspaces of PG(n, q)) is a subgeometry PG(h(n − k)/e, p e), if it exists (recall q = p h , so 1 ≤ e ≤ h and e|h).
Our word posets have finite words of bounded length as their elements, with the words composed fr... more Our word posets have finite words of bounded length as their elements, with the words composed from a finite alphabet. Their partial ordering follows from the inclusion of a word as a subsequence of another word. The elemental combinatorial properties of such posets are established. Their automorphism groups are determined (along with similar result for the word poset studied by Burosch, Frank and Röhl [4]) and a BLYM inequality is verified (via the normalized matching property).
There are many examples for point sets in finite geometry which behave "almost regularly" in some... more There are many examples for point sets in finite geometry which behave "almost regularly" in some (well-defined) sense, for instance they have "almost regular" line-intersection numbers. In this paper we investigate point sets of a desarguesian affine plane, for which there exist some (sometimes: many) parallel classes of lines, such that almost all lines of one parallel class intersect our set in the same number of points (possibly mod p, the characteristic). The lines with exceptional intersection numbers are called renitent, and we prove results on the (regular) behaviour of these renitent lines. As a consequence of our results, we also prove geometric properties of codewords of the F p-linear code generated by characteristic vectors of lines of PG(2, q).
Studia Scientiarum Mathematicarum Hungarica, Dec 1, 2008
The attempts to construct a counterexample to the Strong Perfect Graph Conjecture yielded the not... more The attempts to construct a counterexample to the Strong Perfect Graph Conjecture yielded the notion of partitionable graphs as minimal imperfect graphs; then nearfactorizations of nite groups gained some interest since from any near-factorization some partitionable graphs can be constructed in a natural way. Recently, the proof of SPGC was declared by Chudnovsky, Robertson, Seymour and Thomas [3], but near-factorizations remain interesting on their own rights as (i) rare objects being close to factorizations of groups; and (ii) they yield graphs with surprising properties. Weak Perfect Graph Conjecture (Theorem since 1972 see [7]). The complement of a perfect graph is also perfect. Strong Perfect Graph Conjecture (SPGC)(Theorem since 2002 see [3]). Every Berge graph is perfect.
It is known that in PG(3, q), q > 19, a partial flock of a quadratic cone with q − ε planes, can ... more It is known that in PG(3, q), q > 19, a partial flock of a quadratic cone with q − ε planes, can be extended to a unique flock if ε < 1 4 √ q, and a similar and slightly stronger theorem holds for the case q even. In this paper we prove the analogue of this result for cones with base curve of higher degree.
In this article we prove a theorem about the number of directions determined by less then q affin... more In this article we prove a theorem about the number of directions determined by less then q affine points, similar to the result of Blokhuis et al. (in J. Comb. Theory Ser. A 86(1), 187-196, 1999) on the number of directions determined by q affine points.
In this paper, we show that a small minimal blocking set with exponent e in PG(n, p t), p prime, ... more In this paper, we show that a small minimal blocking set with exponent e in PG(n, p t), p prime, spanning a (t/e − 1)-dimensional space, is an F p e-linear set, provided that p > 5(t/e) − 11. As a corollary, we get that all small minimal blocking sets in PG(n, p t), p prime, p > 5t − 11, spanning a (t − 1)-dimensional space, are F p-linear, hence confirming the linearity conjecture for blocking sets in this particular case.
The journal of combinatorial mathematics and combinatorial computing, 2007
Let m 2 (N, q) denote the size of the largest caps in P G(N, q) and let m ′ 2 (N, q) denote the s... more Let m 2 (N, q) denote the size of the largest caps in P G(N, q) and let m ′ 2 (N, q) denote the size of the second largest complete caps in P G(N, q). Presently, it is known that m 2 (4, 5) ≤ 111 and that m 2 (4, 7) ≤ 316. Via computer searches for caps in P G(4, 5) using the result of Abatangelo, Larato and Korchmáros that m ′ 2 (3, 5) = 20, we improve the first upper bound to m 2 (4, 5) ≤ 88. Computer searches in P G(3, 7) show that m ′ 2 (3, 7) = 32 and this latter result then improves the upper bound on m 2 (4, 7) to m 2 (4, 7) ≤ 238. We also present the known upper bounds on m 2 (N, 5) and m 2 (N, 7) for N > 4.
There is a still growing theory of Rédei type blocking sets and their applications, also of the s... more There is a still growing theory of Rédei type blocking sets and their applications, also of the set of directions determined by the graph of a function. Here we prove a theorem about the number of directions determined by a pointset of size p 2 in AG(3, p), where p is prime. Then two results, which are applications of the planar theorem, are generalized using the new theorem.
We will not be very strict and consistent in the notation (but at least we'll try to be). However... more We will not be very strict and consistent in the notation (but at least we'll try to be). However, here we give a short description of the typical notation we are going to use. If not specified differently, q = p h is a prime power, p is a prime, and we work in the Desarguesian projective (or affine) space PG(n, q) (AG(n, q), resp.), each space coordinatized by the finite (Galois) field GF(q). The n-dimensional vectorspace over GF(q) will be denoted by V(n, q) or simply by GF(q) n. When discussing PG(n, q) and the related V(n + 1, q) together then for a subspace dimension will be meant rank=dim+1 projectively while vector space dimension will be called rank. A field, which is not necessarily finite will be denoted by F. In general capital letters X, Y, Z, T, ... will denote independent variables, while x, y, z, t, ... will be elements of GF(q). A pair or triple of variables or elements in any pair of brackets can be meant homogeneously, hopefully it will be always clear from the context and the actual setting. We write X or V = (X, Y, Z, ..., T) meaning as many variables as needed; V q = (X q , Y q , Z q , ...). As over a finite field of order q for each x ∈ GF(q) x q = x holds, two different polynomials, f and g, in one or more variables, can have coinciding values "everywhere" over GF(q). In this case we ought to write f ≡ g, as for univariate polynomials f (X), g(X) it means that f ≡ g (mod X q − X) in the ring GF(q)[X]. However, as in the literature f ≡ g is used in the sense 'f and g are equal as polynomials', we will use it in the same sense; though simply f = g and f (X) = g(X) may denote the same. Throughout this book we mostly use the usual representation of PG(n, q). This means that the points have homogeneous coordinates (x, y, z, ..., t) where x, y, z, ..., t are elements of GF(q). The hyperplane [a, b, c, ..., d] of the space have equation aX + bY + cZ + ... + dT = 0. When PG(n, q) is considered as AG(n, q) plus the hyperplane at infinity, then we will use the notation H ∞ for that ('ideal') hyperplane. If n = 2 then H ∞ is called the line at infinity ∞. According to the standard terminology, a line meeting a pointset in one point will be called a tangent and a line intersecting it in r points is an r-secant (or a line of length r). This book is about combinatorially defined (point)sets of (mainly projective or affine) finite geometries. They are defined by their intersection numbers with lines (or other subspaces) typically. The most important definitions and basic information are collected in the Glossary of concepts at the end of this book. These are: blocking sets, arcs, nuclei, spreads, sets of even type, etc. Warning. In this book a curve is allowed to have multiple components, so in fact the curves considered here are called cycles in a different terminology.
Given a set T ⊆ GF(q), |T | = t, w T is defined as the smallest positive integer k for which y∈T ... more Given a set T ⊆ GF(q), |T | = t, w T is defined as the smallest positive integer k for which y∈T y k = 0. It can be shown that w T ≤ t always and w T ≤ t − 1 if the characteristic p divides t. T is called a Vandermonde set if w T ≥ t−1 and a super-Vandermonde set if w T = t. This (extremal) algebraic property is interesting for its own right, but the original motivation comes from finite geometries. In this paper we classify small and large super-Vandermonde sets.
In this paper, we associate a weight function with a set of points satisfying the conditions of t... more In this paper, we associate a weight function with a set of points satisfying the conditions of the cylinder conjecture. Then we derive properties of this weight function using the Rédei polynomial of the point set. Using additional assumptions on the number of non-determined directions, together with an exhaustive computer search for weight functions satisfying particular properties, we prove a relaxed version of the cylinder conjecture for $$p \le 13$$p≤13. This result also slightly refines a result of Sziklai on point sets in $$\mathrm {AG}(3,p)$$AG(3,p).
Our word posets have finite words of bounded length as their elements, with the words composed fr... more Our word posets have finite words of bounded length as their elements, with the words composed from a finite alphabet. Their partial ordering follows from the inclusion of a word as a subsequence of another word. The elemental combinatorial properties of such posets are established. Their automorphism groups are determined (along with similar result for the word poset studied by Burosch, Frank and Röhl [4]) and a BLYM inequality is verified (via the normalized matching property).
Van Lint and MacWilliams (IEEE Trans. Inform. Theory IT 24 (1978) 730-737) conjectured that the o... more Van Lint and MacWilliams (IEEE Trans. Inform. Theory IT 24 (1978) 730-737) conjectured that the only q-subset X of GF(q 2), with the properties 0; 1 ∈ X and x − y is a square for all x; y ∈ X , is the set GF(q). Aart Blokhuis (Indag. Math. 46 (1984) 369-372) proved the conjecture for arbitrary odd q. In this paper we give a similar characterization of GF(q) in GF(q 2), proving the analogue of Blokhuis' theorem for dth powers (instead of squares), when d|(q + 1). We also prove an embedding-type result, stating that if |S| ¿ q − (1 − 1=d) √ q with the same properties as X above, then S ⊆ GF(q).
In [10], it was shown that small t-fold (n − k)-blocking sets in PG(n, q), q = p h , p prime, h ≥... more In [10], it was shown that small t-fold (n − k)-blocking sets in PG(n, q), q = p h , p prime, h ≥ 1, intersect every k-dimensional space in t (mod p) points. We characterize in this article all t-fold (n − k)-blocking sets in PG(n, q), q square, q ≥ 661, t < c p q 1/6 /2, |B| < tq n−k + 2tq n−k−1 √ q, intersecting every k-dimensional space in t (mod √ q) points. * The third author is grateful for the partial support of OTKA T049662, T067867 and Bolyai grants. In the theory of 1-fold planar blocking sets, 1 (mod p) results for small 1-fold planar blocking sets play an important role. Definition 1.3 A blocking set of PG(2, q) is called small when it has less than 3(q + 1)/2 points. If q = p h , p prime, h ≥ 1, the exponent e of the minimal blocking set B of PG(2, q) is the maximal integer e such that every line intersects B in 1 (mod p e) points. Theorem 1.4 Let B be a small minimal 1-fold blocking set in PG(2, q), q = p h , p prime, h ≥ 1. Then B intersects every line in 1 (mod p) points, so for the exponent e of B, we have 1 ≤ e ≤ h. (Szőnyi [18]) In fact, this exponent e is a divisor of h. (Sziklai [17]) This result was extended by Szőnyi and Weiner [19] to 1-fold (n − k)blocking sets in PG(n, q). Definition 1.5 A 1-fold (n−k)-blocking set of PG(n, q) is called small when it has less than 3(q n−k + 1)/2 points. If q = p h , p prime, h ≥ 1, the exponent e of the minimal 1-fold (n − k)blocking set B is the maximal integer e such that every hyperplane intersects B in 1 (mod p e) points. A most interesting question of the theory of blocking sets is to classify the small blocking sets. A natural construction (blocking the k-subspaces of PG(n, q)) is a subgeometry PG(h(n − k)/e, p e), if it exists (recall q = p h , so 1 ≤ e ≤ h and e|h).
Our word posets have finite words of bounded length as their elements, with the words composed fr... more Our word posets have finite words of bounded length as their elements, with the words composed from a finite alphabet. Their partial ordering follows from the inclusion of a word as a subsequence of another word. The elemental combinatorial properties of such posets are established. Their automorphism groups are determined (along with similar result for the word poset studied by Burosch, Frank and Röhl [4]) and a BLYM inequality is verified (via the normalized matching property).
There are many examples for point sets in finite geometry which behave "almost regularly" in some... more There are many examples for point sets in finite geometry which behave "almost regularly" in some (well-defined) sense, for instance they have "almost regular" line-intersection numbers. In this paper we investigate point sets of a desarguesian affine plane, for which there exist some (sometimes: many) parallel classes of lines, such that almost all lines of one parallel class intersect our set in the same number of points (possibly mod p, the characteristic). The lines with exceptional intersection numbers are called renitent, and we prove results on the (regular) behaviour of these renitent lines. As a consequence of our results, we also prove geometric properties of codewords of the F p-linear code generated by characteristic vectors of lines of PG(2, q).
Studia Scientiarum Mathematicarum Hungarica, Dec 1, 2008
The attempts to construct a counterexample to the Strong Perfect Graph Conjecture yielded the not... more The attempts to construct a counterexample to the Strong Perfect Graph Conjecture yielded the notion of partitionable graphs as minimal imperfect graphs; then nearfactorizations of nite groups gained some interest since from any near-factorization some partitionable graphs can be constructed in a natural way. Recently, the proof of SPGC was declared by Chudnovsky, Robertson, Seymour and Thomas [3], but near-factorizations remain interesting on their own rights as (i) rare objects being close to factorizations of groups; and (ii) they yield graphs with surprising properties. Weak Perfect Graph Conjecture (Theorem since 1972 see [7]). The complement of a perfect graph is also perfect. Strong Perfect Graph Conjecture (SPGC)(Theorem since 2002 see [3]). Every Berge graph is perfect.
It is known that in PG(3, q), q > 19, a partial flock of a quadratic cone with q − ε planes, can ... more It is known that in PG(3, q), q > 19, a partial flock of a quadratic cone with q − ε planes, can be extended to a unique flock if ε < 1 4 √ q, and a similar and slightly stronger theorem holds for the case q even. In this paper we prove the analogue of this result for cones with base curve of higher degree.
In this article we prove a theorem about the number of directions determined by less then q affin... more In this article we prove a theorem about the number of directions determined by less then q affine points, similar to the result of Blokhuis et al. (in J. Comb. Theory Ser. A 86(1), 187-196, 1999) on the number of directions determined by q affine points.
In this paper, we show that a small minimal blocking set with exponent e in PG(n, p t), p prime, ... more In this paper, we show that a small minimal blocking set with exponent e in PG(n, p t), p prime, spanning a (t/e − 1)-dimensional space, is an F p e-linear set, provided that p > 5(t/e) − 11. As a corollary, we get that all small minimal blocking sets in PG(n, p t), p prime, p > 5t − 11, spanning a (t − 1)-dimensional space, are F p-linear, hence confirming the linearity conjecture for blocking sets in this particular case.
The journal of combinatorial mathematics and combinatorial computing, 2007
Let m 2 (N, q) denote the size of the largest caps in P G(N, q) and let m ′ 2 (N, q) denote the s... more Let m 2 (N, q) denote the size of the largest caps in P G(N, q) and let m ′ 2 (N, q) denote the size of the second largest complete caps in P G(N, q). Presently, it is known that m 2 (4, 5) ≤ 111 and that m 2 (4, 7) ≤ 316. Via computer searches for caps in P G(4, 5) using the result of Abatangelo, Larato and Korchmáros that m ′ 2 (3, 5) = 20, we improve the first upper bound to m 2 (4, 5) ≤ 88. Computer searches in P G(3, 7) show that m ′ 2 (3, 7) = 32 and this latter result then improves the upper bound on m 2 (4, 7) to m 2 (4, 7) ≤ 238. We also present the known upper bounds on m 2 (N, 5) and m 2 (N, 7) for N > 4.
There is a still growing theory of Rédei type blocking sets and their applications, also of the s... more There is a still growing theory of Rédei type blocking sets and their applications, also of the set of directions determined by the graph of a function. Here we prove a theorem about the number of directions determined by a pointset of size p 2 in AG(3, p), where p is prime. Then two results, which are applications of the planar theorem, are generalized using the new theorem.
We will not be very strict and consistent in the notation (but at least we'll try to be). However... more We will not be very strict and consistent in the notation (but at least we'll try to be). However, here we give a short description of the typical notation we are going to use. If not specified differently, q = p h is a prime power, p is a prime, and we work in the Desarguesian projective (or affine) space PG(n, q) (AG(n, q), resp.), each space coordinatized by the finite (Galois) field GF(q). The n-dimensional vectorspace over GF(q) will be denoted by V(n, q) or simply by GF(q) n. When discussing PG(n, q) and the related V(n + 1, q) together then for a subspace dimension will be meant rank=dim+1 projectively while vector space dimension will be called rank. A field, which is not necessarily finite will be denoted by F. In general capital letters X, Y, Z, T, ... will denote independent variables, while x, y, z, t, ... will be elements of GF(q). A pair or triple of variables or elements in any pair of brackets can be meant homogeneously, hopefully it will be always clear from the context and the actual setting. We write X or V = (X, Y, Z, ..., T) meaning as many variables as needed; V q = (X q , Y q , Z q , ...). As over a finite field of order q for each x ∈ GF(q) x q = x holds, two different polynomials, f and g, in one or more variables, can have coinciding values "everywhere" over GF(q). In this case we ought to write f ≡ g, as for univariate polynomials f (X), g(X) it means that f ≡ g (mod X q − X) in the ring GF(q)[X]. However, as in the literature f ≡ g is used in the sense 'f and g are equal as polynomials', we will use it in the same sense; though simply f = g and f (X) = g(X) may denote the same. Throughout this book we mostly use the usual representation of PG(n, q). This means that the points have homogeneous coordinates (x, y, z, ..., t) where x, y, z, ..., t are elements of GF(q). The hyperplane [a, b, c, ..., d] of the space have equation aX + bY + cZ + ... + dT = 0. When PG(n, q) is considered as AG(n, q) plus the hyperplane at infinity, then we will use the notation H ∞ for that ('ideal') hyperplane. If n = 2 then H ∞ is called the line at infinity ∞. According to the standard terminology, a line meeting a pointset in one point will be called a tangent and a line intersecting it in r points is an r-secant (or a line of length r). This book is about combinatorially defined (point)sets of (mainly projective or affine) finite geometries. They are defined by their intersection numbers with lines (or other subspaces) typically. The most important definitions and basic information are collected in the Glossary of concepts at the end of this book. These are: blocking sets, arcs, nuclei, spreads, sets of even type, etc. Warning. In this book a curve is allowed to have multiple components, so in fact the curves considered here are called cycles in a different terminology.
Given a set T ⊆ GF(q), |T | = t, w T is defined as the smallest positive integer k for which y∈T ... more Given a set T ⊆ GF(q), |T | = t, w T is defined as the smallest positive integer k for which y∈T y k = 0. It can be shown that w T ≤ t always and w T ≤ t − 1 if the characteristic p divides t. T is called a Vandermonde set if w T ≥ t−1 and a super-Vandermonde set if w T = t. This (extremal) algebraic property is interesting for its own right, but the original motivation comes from finite geometries. In this paper we classify small and large super-Vandermonde sets.
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