Algebrai módszerek a Galois-geometriában = Algebraic methods in Galois-geometries

Paros q-ra stabilitasi eredmenyt bizonyitottunk PG(2,q) paros halmazaira. Ez negyzet q-ra eles, es B. Segre ivek beagyazasarol szolo hires tetelet altalanositja. Megmutattuk, hogy negyzet q-ra PG(2,q)-ban 4qlog q es q^(3/2)-q+2q^(1/2) kozott minden meretű minimalis lefogo ponthalmaz letezik, sőt egy kicsit szűkebb intervallum minden ertekere q-ban tobb, mint polinomnyi. Magasabb dimenzios projektiv terekben a hipersikokat r modulo p pontban metsző halmazok meretere bizonyos esetekben eles also becslest adtunk, amely a maximalis ivek nemletezesere vonatkozo Ball-Blokhuis-Mazzocca tetel altalanositasa. Ez oszthato linearis kodok hosszara az n legalabb (r-1)q+(p-1)r also becslest adja, ahol r az az ertek, amellyel n es minden kodszo sulya is oszthato. Megmutattuk, hogy PG(2,q) regularis szemiovalisai csak az ovalisok es az unitalok. Segre tipusu eredmenyt sikerult belatni masod es magasabbrendű kupok reszleges kupszeletnyalabjaira. Kis minimalis lefogo ponthalmazok strukturajarol azt sikerult megmutatni, hogy ezek minden egyenest 1 modulo p^e pontban metszenek, ahol e osztja h-t, ha q=p^h. Ezen tulmenően, ha a metszet p^e+1 elemű, akkor az GF(p^e) feletti reszegyenes. Kis t-szeres lefogo ponthalmazokra az egyenesekkel valo metszetekre belattuk, hogy azok modulo p t-vel kongruensek, ahol t a karakterisztika. Megmutattuk, hogy a Q(4,q) altalanositott negyszogben nincsenek q^2-1 pontu maximalis parcialis ovoidok. PG(3m-1,q) sikokkal valo reszleges befedeseire adtunk konstrukciokat. | For even q-s we proved a stability theorem for sets of even type in PG(2,q). The result is sharp when q is a square, and it generalizes a famous embeddability theorem for arcs, due to B. Segre. It was proven that in PG(2,q), q square, there is a minimal blocking set for any size between 4qlog q and q^(3/2)-q+2q^(1/2), Moreover, for a slightly smaller interval we also proved that the number of nonisomorphic minimal blocking sets of that size is more than polynomial in q. For sets intersecting all hyperplanes in r modulo p points we found a lower bound that is sharp in some cases. The proof generalizes the nonexistence of maximal arcs, due to Ball-Blokhuis-Mazzocca. For divisible linear codes it gives that the length is at least (r-1)q+(p-1)r, where divides the length and the weight of all codewords. We found that in PG(2,q) regular semiovals must be either ovals or unitals. We obtained a Segre type theorem for partial flocks of the quadratic and general cones. About the structure of small minimal blocking sets we obtained the following: each line intersects the set in 1 modulo p^e points, where e divides h and q=p^h. Furthermore, if the intersection has p^e+1 points, then it is a subline over GF(p^e). We proved that a small minimal t-fold blocking set intersects every line in t modulo p points, where p is the characteristics. We also proved that the GQ Q(4,q) does not have maximal partial ovoids of size q^2-1. We gave constructions for partial plane spreads of PG(3m-1,q).