By Michael Huber
As a result of the category of the finite uncomplicated teams, it hasbeen attainable in recent times to represent Steiner t-designs, that's t -(v, okay, 1) designs,mainly for t = 2, admitting teams of automorphisms with sufficiently strongsymmetry houses. in spite of the fact that, regardless of the finite easy workforce type, forSteiner t-designs with t > 2 almost all these characterizations have remained longstandingchallenging difficulties. specially, the decision of all flag-transitiveSteiner t-designs with three ≤ t ≤ 6 is of specific curiosity and has been open for about40 years (cf. Delandtsheer (Geom. Dedicata forty-one, p. 147, 1992 and guide of IncidenceGeometry, Elsevier technological know-how, Amsterdam, 1995, p. 273), yet possibly datingback to 1965).The current paper maintains the author's paintings (see Huber (J. Comb. thought Ser.A ninety four, 180-190, 2001; Adv. Geom. five, 195-221, 2005; J. Algebr. Comb., 2007, toappear)) of classifying all flag-transitive Steiner 3-designs and 4-designs. We provide acomplete class of all flag-transitive Steiner 5-designs and turn out furthermorethat there are not any non-trivial flag-transitive Steiner 6-designs. either effects depend on theclassification of the finite 3-homogeneous permutation teams. additionally, we surveysome of the main common effects on hugely symmetric Steiner t-designs.
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Extra resources for A census of highly symmetric combinatorial designs
Note that in practice, n will be the relative degree of number fields extensions, and so in many cases the naive algorithm will be sufficient. 6. We first need a definition. 8. Let (A, I) be a pseudo-matrix with I = (aj ) · If i 1 , . . , i r are r distinct rows of A and ii , . . , ir are r distinct columns, we define the minor-ideal corresponding to these indices as follows. Let d be the determinant of the r x r minor extracted from the given rows and columns of A. Then the minor-ideal is the ideal daj1 • • • air .
In patticular, if so desired, we may assume that the ai are integral ideals, or that the Wi are elements of M. On the other hand, it is generally not possible to have both properties at once. For example, let M = a be a nonprincipal, primitive integral ideal. The general pseudo-basis of M is ( a , a/ a ) , and so to have both an element of M and an integral ideal, we would need a E a and a/ a C R, which is equivalent to a = aR, contrary to our choice of a. Furthermore, restricting either to elements of M or to integral ideals would be too rigid for algorithmic purposes, so it is preferable not to choose a pseudo-basis of a particular type.
We write this out explicitly as an algorithm. 4. 1 2 (HNF Reduction Modulo an Ideal) . Given an idea l a by its m x m u pper-tria ngular H N F matrix H = (hi,j) in some basis of K, and an element x E K given by a column vector X = (xi ) in the same basis, this a lgorith m computes a "canonical" representative of x modulo a, more precisely an element y E K such that x - y E a and the coordinates Yi of y in the basis satisfy 0 s Yi < hi,i · 1 . [Initialize) Set i +- m, y +- x. 2. [Reduce) Set q +- lyi / hi,i j , y +- y - q Hi (recall that Hi i s the ith column of H) .
A census of highly symmetric combinatorial designs by Michael Huber