Permutations, hyperplanes and polynomials over finite fields

Starting with a result in combinatorial number theory we prove that (apart from a couple of exceptions that can be classified). for any elements a1, .,a(n) of GF(q), there are distinct field elements a(1), a(n), such that a(1)b(1) + +a(n)b(n) = 0. This implies the classification of hyperplanes lying in the union of the hyperplanes X-i = X-j in a vector space over GF(q), and also the classification of those multisets for which all reduced polynomials of this range are of reduced degree q - 2. The proof is based on the polynomial method. (C) 2010 Elsevier Inc. All rights reserve

Permutations over cyclic groups

Generalizing a result in the theory of finite fields we prove that, apart from a couple of exceptions that can be classified, for any elements $a_1,...,a_m$ of the cyclic group of order $m$, there is a permutation $\pi$ such that $1a_{\pi(1)}+...+ma_{\pi(m)}=0$

Kódelmélet és környéke = Coding theory and its neighbourhood

Kódelméletben hasznos véges projektív terek speciális egyenes-, illetve hipersíkmetszetű ponthalmazainak vizsgálata. Egyes cikkeinknek közvetlen kódelméleti alkalmazása van (itt ezeket soroljuk), mások geometriai ill. algebrai szálakkal kapcsolódnak oda. A polinomos módszer alkalmazásával bebizonyítottuk, hogy PG(2,q) egy olyan ponthalmaza, melyet minden egyenes adott r mod p pontban metsz, legalább (r-1)q+(p-1)r pontú kell legyen, ahol p a karakterisztika, r|q. Következésképp egy 3 dimenziós kód,melynek hossza és súlyai is oszthatók r-rel és minimális távolsága legalább 3, legalább (r-1)q+(p-1)r hosszú kell legyen. Ball, Blokhuis és Mazzocca híres, maximális ívek nemlétezéséről szóló tétele is egyszerűen kijön a tételből. Meghatároztuk két fontos poset, D^{k,n} és B_{m,n} automorfizmus-csoportját.A kérdéskör az insertion-deletion kódokhoz kapcsolódik. A B_{m,n} struktúra automorfizmus-csoportja korábban is ismert volt, de a hosszú bizonyítást 1 oldalasra redukáltuk. Megfogalmaztunk egy sejtést algebrai síkgörbék pontjainak számáról: n-edfokú, lineáris komponens nélküli görbének legfeljebb (n-1)q+1 pontja lehet; (n-1)q+n/2-t sikerült igazolni. Ilyen görbék hatékony kódokat adnak. Bebizonyítottuk, hogy ha egy lineáris [n,k,d]_q kód kiterjeszthető nem feltétlenül lineáris [n+1,k,d+1]_q kóddá, akkor a kiterjesztést lineáris módon is meg lehet csinálni. Eredményünk kiterjesztéséből pedig az következhetne,hogy az MDS sejtés lineáris és tetszőleges kódokra ekvivalens. | In coding theory, it is useful to study point sets of finite projective spaces with special intersection multiplicities with respect to lines and hyperplanes. Some of our papers have immediate application in coding theory (here we list those), the others are linked by its geometrical or algebraical concept. Using polynomial method, we proved that point sets of PG(n,q) intersecting each hyperplane in r mod p points have at least (r-1)q+(p-1)r points, where p is the characteristic and r|q. Hence a linear code whose length and weight are divisible by r and whose dual minimum distance is at least 3, has length at least (r-1)q+(p-1)r. Now the famous Ball-Blokhuis-Mazzocca theorem on the non-existence of maximal arcs becomes a corollary of this result. We determined the automorphism group of two important posets D^{k,n} and B_{m,n}. It was already known for B_{m,n}, but we shortened its long proof to 1 page. This topic is related to insertion-deletion codes. We conjecture that an algebraic plane curve of degree n without linear components can have at most (n-1)q+1 points, we showed that it is at most (n-1)q+n/2. Such curves give efficient codes. We proved that if a linear [n,k,d]_q code can be extended to a not necessarily linear[n+1,k,d+1]_q code then it can be done also in a linear way. From an extension of our results it would follow that the MDS-conjecture is equivalent for linear and arbirtary codes

Decomposition of Geometric Set Systems and Graphs

We study two decomposition problems in combinatorial geometry. The first part deals with the decomposition of multiple coverings of the plane. We say that a planar set is cover-decomposable if there is a constant m such that any m-fold covering of the plane with its translates is decomposable into two disjoint coverings of the whole plane. Pach conjectured that every convex set is cover-decomposable. We verify his conjecture for polygons. Moreover, if m is large enough, we prove that any m-fold covering can even be decomposed into k coverings. Then we show that the situation is exactly the opposite in 3 dimensions, for any polyhedron and any $m$ we construct an m-fold covering of the space that is not decomposable. We also give constructions that show that concave polygons are usually not cover-decomposable. We start the first part with a detailed survey of all results on the cover-decomposability of polygons. The second part investigates another geometric partition problem, related to planar representation of graphs. The slope number of a graph G is the smallest number s with the property that G has a straight-line drawing with edges of at most s distinct slopes and with no bends. We examine the slope number of bounded degree graphs. Our main results are that if the maximum degree is at least 5, then the slope number tends to infinity as the number of vertices grows but every graph with maximum degree at most 3 can be embedded with only five slopes. We also prove that such an embedding exists for the related notion called slope parameter. Finally, we study the planar slope number, defined only for planar graphs as the smallest number s with the property that the graph has a straight-line drawing in the plane without any crossings such that the edges are segments of only s distinct slopes. We show that the planar slope number of planar graphs with bounded degree is bounded.Comment: This is my PhD thesi