Semifields in loop theory and in finite geometry

This paper is a relatively short survey the aim of which is to present the theory of semifields and the related areas of finite geometry to loop theorists

Diszkrét geometria és geometriai algebra = Discrete geometry and geometric algebra

Nagy G. olyan geometriai struktúrákat vizsgált, melyek Moufang-féle és Bol-féle egységelemes kvázicsoportokkal (loopokkal) koordinátázhatóak. a) Kis Frattini 2-loopok, azaz melyeknél L/A elemi Abel 2-csoport valamely 2-rendű A normális részloopra. A Bol-esetben explicit formulát, a Moufang-esetben új globális konstrukciót adott. b) Moufang-féle p-loopok, p>3. Ilyen loopokra korábban nagyon kevés példa volt ismert. Nagy G. M. Valsecchivel fontos azonosságokat talált nilpotens Moufang-loopokra és egy általános új konstrukciót talált, továbbá osztályozták a p^5 (p>3) rendű Moufang-loopokat is. c) Kis Moufang- és Bol-loopok osztályozása, a P. Vojtechovsky-val közösen készített komputeralgebrai programcsomag felhasználásával. Fodor F. megtalálta 13 és 14 egybevágó kör körbe való legsűrűbb elhelyezéseit. Ambrus G.-vel közösen Fodor F. új alsó korlátot bizonyított 3-dimenziós egységgömb elhelyezésekbeli Voronoi cellák felszínére. T. Bisztriczkyvel és D. Oliverosszal közösen Fodor F. bebizonyította, hogy ha egy páronként diszjunkt körökből álló rendszerben minden 4-elemű részhalmaznak van transzverzálisa, akkor van olyan egyenes, ami legfeljebb egy kivételével a rendszer minden elemét metszi. Ambrus G.-vel és Bezdek A.-val közösen Fodor F. megmutatta, hogy ha egy n-dimenziós egységgömbökből álló rendszerben, ahol a középpontok távolsága legalább 3.6955..., minden n^2-elemű részhalmaznak van transzverzálisa, akkor az egész rendszernek is van transzverzálisa. Fodor F. W. Kuperberggel és T. Bisztriczkyvel közösen ""Discrete Geometry"" című konferenciakötetet szerkesztett. | G. Nagy studied geometric structures which can be coordinatized by Moufang and Bol loops. a) Small Frattini 2-loops are loops L with a normal subloop A of order 2 such that L/A is an elementary Abelian 2-group. Nagy gave an explicit formula in the Bol case and a new global construction in the Moufang case. b) Moufang p-loops with p>3. Before, there were not many examples known for such loops. Together with M. Valsecchi, G. Nagy found some important identities for this class of loops. Using these, they gave a very general new construction and classified all Moufang loops of order p^5 for p>3. c) Jointly with P. Vojtechovsky, G. Nagy wrote a computer algebra package for loops. They used this package to classify small Moufang and Bol loops. F. Fodor found the densest packings of 13 and 14 congruent circles in a circle. Jointly with G. Ambrus, F. Fodor proved a new lower bound for the surface area of Voronoi polyhedra in 3-dimensional unit ball packings. With T. Bisztriczky and D. Oliveros, F. Fodor proved that if in a family of pairwise disjoint unit disks every 4-membered subfamily has a transversal line, then there is a line that intersects all members of the family with the possible exception of at most one. Jointly with G. Ambrus and A. Bezdek, F. Fodor showed that if in a family of n-dimensional unit balls in which the centres of the balls are at least 3.6955... apart every n^2-membered subfamily has a transversal, then the whole family has a transversal. F. Fodor co-edited a conference proceedings volume ""Discrete Geometry"" with T. Bisztriczky and W. Kuperberg

New Steiner 2-designs from old ones by paramodifications

Techniques of producing new combinatorial structures from old ones are commonly called trades. The switching principle applies for a broad class of designs: it is a local transformation that modifies two columns of the incidence matrix. In this paper, we present a construction, which is a generalization of the switching transform for the class of Steiner 2-designs. We call this construction paramodification of Steiner 2-designs, since it modifies the parallelism of a subsystem. We study in more detail the paramodifications of affine planes, Steiner triple systems, and abstract unitals. Computational results show that paramodification can construct many new unitals

On the nonexistence of certain orthogonal arrays of strength four

We show that no orthogonal arrays OA(16A, 11,2,4) exist with A = 6 and 7. This solves an open problem of the NSUCRYPTO Olympiad 2018. Our result allows to determine the minimum weights of certain higher order correlation-immune Boolean functions

Estimating the dimension of the subfield subcodes of hermitian codes

In this paper, we study the behavior of the true dimension of the subfield subcodes of Hermitian codes. Our motivation is to use these classes of linear codes to improve the parameters of the McEliece cryptosystem, such as key size and security level. The McEliece scheme is one of the promising alternative cryptographic schemes to the current public key schemes since in the last four decades, they resisted all known quantum computing attacks. By computing and analyzing a data collection of true dimensions of subfield subcodes, we concluded that they can be estimated by the extreme value distribution function

Selection criteria for preoperative endoscopic retrograde cholangiopancreatography before laparoscopic cholecystectomy and endoscopic treatment of bile duct stones. Results of a retrospective; single center study between 1996-2002

AIM: The optimal treatment for bile duct stones (in terms of cost, complications and accuracy) is unclear. The aim of our study was to determine the predictive factors for preoperative endoscopic retrograde cholangiopancreatography (ERCP). METHODS: Patients undergoing preoperative ERCP (= 8 mm) and/or stone at US examination, coexisting acute pancreatitis and/or acute pancreatitis or jaundice in patient's history. Suspected prognostic factors and the combination of factors were compared to the result of ERCP. RESULTS: Two hundred and six preoperative ERCPs were performed during the observed period. The rate of successful cannulation for ERC was (97.1%). Bile duct stones were detected in 81 patients (39.3%), and successfully removed in 79 (97.5%). The number of prognostic factors correlated with the presence of bile duct stones. The positive predictive value for one prognostic factor was 1.2%, for two 43%, for three 72.5%, for four or more 91.4%. CONCLUSION: Based on our data preoperative ERCP is highly recommended in patients with three or more positive factors (high risk patients). In contrast, ERCP is not indicated in patients with zero or one factor (low risk patients). Preoperative ERCP should be offered to patients with two positive factors (moderate risk patients), however the practice should also be based on the local conditions (e.g. skill of the endoscopist, other diagnostic tools)