We construct and study a new near octagon of order (2, 10) which has its full automorphism group isomorphic to the group G 2 (4):2 and which contains 416 copies of the Hall-Janko near octagon as full subgeometries. Using this near octagon and its substructures we give geometric constructions of the G 2 (4)-graph and the Suzuki graph, both of which are strongly regular graphs contained in the Suzuki tower. As a subgeometry of this octagon we have discovered another new near octagon, whose order is (2, 4).

The research in this paper was motivated by one of the most important open problems in the theory of generalized polygons, namely the existence problem for semi-finite thick generalized polygons. We show here that no semi-finite generalized hexagon of order (2, t) can have a subhexagon H of order 2. Such a subhexagon is necessarily isomorphic to the split Cayley generalized hexagon H(2) or its point-line dual H D (2). In fact, the employed techniques allow us to prove a stronger result. We show that every near hexagon S of order (2, t) which contains a generalized hexagon H of order 2 as an isometrically embedded subgeometry must be finite. Moreover, if H ∼ = H D (2) then S must also be a generalized hexagon, and consequently isomorphic to either H D (2) or the dual twisted triality hexagon T (2, 8).

In recent work, we constructed a new near octagon G from certain involutions of the finite simple group G 2 (4) and showed a correspondence between the Suzuki tower of finite simple groups, L 3 (2) < U 3 (3) < J 2 < G 2 (4) < Suz, and the tower of near polygons,Here we characterize each of these near polygons (except for the first one) as the unique near polygon of the given order and diameter containing an isometrically embedded copy of the previous near polygon of the tower. In particular, our characterization of the Hall-Janko near octagon HJ is similar to an earlier characterization due to Cohen and Tits who proved that it is the unique regular near octagon with parameters (2, 4; 0, 3), but instead of regularity we assume existence of an isometrically embedded dual split Cayley hexagon, H(2) D . We also give a complete classification of near hexagons of order (2, 2) and use it to prove the uniqueness result for H(2) D .

A 1993 result of Alon and Füredi gives a sharp upper bound on the number of zeros of a multivariate polynomial over an integral domain in a finite grid, in terms of the degree of the polynomial. This result was recently generalized to polynomials over an arbitrary commutative ring, assuming a certain "Condition (D)" on the grid which holds vacuously when the ring is a domain. In the first half of this paper we give a further Generalized Alon-Füredi Theorem which provides a sharp upper bound when the degrees of the polynomial in each variable are also taken into account. This yields in particular a new proof of Alon-Füredi. We then discuss the relationship between Alon-Füredi and results of DeMillo-Lipton, Schwartz and Zippel. A direct coding theoretic interpretation of Alon-Füredi Theorem and its generalization in terms of Reed-Muller type affine variety codes is shown which gives us the minimum Hamming distance of these codes. Then we apply the Alon-Füredi Theorem to quickly recover -and sometimes strengthen -old and new results in finite geometry, including the Jamison/Brouwer-Schrijver bound on affine blocking sets. We end with a discussion of multiplicity enhancements.

A construction of Alon and Krivelevich gives highly pseudorandom K k -free graphs on n vertices with edge density equal to Θ(n −1/(k−2) ). In this short note we improve their result by constructing an infinite family of highly pseudorandom K k -free graphs with a higher edge density of Θ(n