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A note on sets of type (3, n) in PG(3, q)
No full textWe prove that if K is a set of type (3, n) with n?> 3 in PG(3, q), then either K is the point-set of a plane in PG(3, 2) or n = q + 3 or (q, n) = (8, 7)
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No full textWe prove that in PG(3, q), a (q^2 + q + 1) -set K of type (m, m + q, m + 2q)_2 such that any external line is contained in exactly one m-secant plane is an oval cone. Hence, if q is odd, then K is a quadratic cone
The 1-rotational Kirkman triple systems of order 33
No full textWe establish that there are exactly 500 KTS(33)s admitting an automorphism group fixing one point and acting regularly on the remainder; 436 are over the cyclic group while 64 are over the dicyclic group. There are exactly 243 nonisomorphic STS(33)s underlying the above KTS(33)s; 211 are over the cyclic group while 32 are over the dicyclic group. This gives a significant improvement on the number of known KTS(33)s (at least 528 instead of at least 28)
On singular 1-rotational Steiner 2-designs
No full textA Steiner 2-design is 1-rotational over a group G if it admits G as an automorphism group fixing one point and acting regularly on the remainder. 1-rotational Steiner 2-designs have come into fashion since 1981, when Phelps and Rosa (Discrete Math. 33 (1981), 57-66) studied Steiner triple systems that are 1-rotational over the cyclic group. While all 1-rotational Steiner 2-designs constructed in the past have exactly one short block-orbit, in this paper we also consider 1-rotational Steiner 2-designs not having this property. We call them singular and we show that they are quite rare. In particular, we enumerate all the abelian 1-rotational 2-(49, 4, 1) designs
The 1-rotational (52,4,1)-RBIBD's
No full textUp to isomorphisms, there are exactly 22 1-rotational resolved (52, 4, 1)-BIBD's
On singular 1-rotational Steiner 2-designs
No full textA Steiner 2-design is 1-rotational over a group G if it admits G as an automorphism group fixing one point and acting regularly on the remainder. 1-rotational Steiner 2-designs have come into fashion since 1981, when Phelps and Rosa (Discrete Math. 33 (1981), 57-66) studied Steiner triple systems that are 1-rotational over the cyclic group. While all 1-rotational Steiner 2-designs constructed in the past have exactly one short block-orbit, in this paper we also consider 1-rotational Steiner 2-designs not having this property. We call them singular and we show that they are quite rare. In particular, we enumerate all the abelian 1-rotational 2-(49, 4, 1) designs
The 1-rotational Kirkman triple systems of order 33
No full textWe establish that there are exactly 500 KTS(33)s admitting an automorphism group fixing one point and acting regularly on the remainder; 436 are over the cyclic group while 64 are over the dicyclic group. There are exactly 243 nonisomorphic STS(33)s underlying the above KTS(33)s; 211 are over the cyclic group while 32 are over the dicyclic group. This gives a significant improvement on the number of known KTS(33)s (at least 528 instead of at least 28)
Some observations on three classical BIBDs constructions
No full textAn examination of three well-known composition techniques is used to obtain some new information about the number of non-isomorphic BIBDs with suitable parameters
A combinatorial characterization of the Baer and the unital cone in PG(3 , q2)
No full textBruen, see [5], and Bruen and Thas, see [7], proved that in PG(2, q^2) a blocking set of type (1, q+ 1)_1 is either a Baer subplane or a unital. In this paper a cone-generalization of this result in PG(3, q^2) is provided
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