A Recursive Construction for New Symmetric Designs

We introduce a recursive construction of regular Handamard matrices with row sum 2h for h=±3ⁿ. Whenever q=(2h – 1)² is a prime power, we construct, for every positive integer m, a symmetric designs with parameters (4h²(q^m+1 – 1)/(q – 1), (2h² – h)q^m, (h² – h)q^m).     

On generalized Hadamard matrices of minimum rank

Generalized Hadamard matrices of order qⁿ⁻¹ (q—a prime power, n2) over GF(q) are related to symmetric nets in affine 2-(qⁿ,qⁿ⁻¹,(qⁿ⁻¹−1)/(q−1)) designs invariant under an elementary abelian group of order q acting semi-regularly on points and blocks. The rank of any such matrix over GF(q) is greater than or equal to n−1. It is proved that a matrix of minimum q-rank is unique up to a monomial equivalence, and the related symmetric net is a classical net in the n-dimensional affine geometry AG(n,q).     

Some designs which admit strong tactical decompositions

Whenever there exist affine planes of orders n ? 1 and n, a construction is given for a 2 ? ((n + 1)(n ? 1)², n(n ? 1), n) design admitting a strong tactical decomposition. These designs are neither symmetric nor strongly resolvable but can be embedded in symmetric 2 ? (n³ ? n + 1, n², n) designs.     

Exponential bounds on the number of designs with affine parameters

It is well‐known that the number of designs with the parameters of a classical design having as blocks the hyperplanes in PG(n, q) or AG(n, q), n≥3, grows exponentially. This result was extended recently [D. Jungnickel, V. D. Tonchev, Des Codes Cryptogr, published online: 23 May, 2009] to designs having the same parameters as a projective geometry design whose blocks are the d‐subspaces of PG(n, q), for any 2≤d≤n−1. In this paper, exponential lower bounds are proved on the number of non‐isomorphic designs having the same parameters as an affine geometry design whose blocks are the d‐subspaces of AG(n, q), for any 2≤d≤n−1, as well as resolvable designs with these parameters. An exponential lower bound is also proved for the number of non‐isomorphic resolvable 3‐designs with the same parameters as an affine geometry design whose blocks are the d‐subspaces of AG(n, 2), for any 2≤d≤n−1. © 2010 Wiley Periodicals, Inc. J Combin Designs 18: 475–487, 2010     

Automorphisms and Isomorphisms of Symmetric and Affine Designs

Given a finite group G, for all sufficiently large d and for each q > 3 there are symmetric designs and affine designs having the same parameters as PG(d, q) and AG(d, q), respectively, and having full automorphism group isomorphic to G.     

Idempotent structure of Clifford algebras

Spinor spaces can be represented as minimal left ideals of Clifford algebras and they are generated by primitive idempotents. Primitive idempotents of the Clifford algebras R _{p, q} are shown to be products of mutually nonannihilating commuting idempotent % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabaGaaiaacaqabeaadaqaaqGaaO% qaamaaleaaleaacaaIXaaabaGaaGOmaaaaaaa!3DBD!\[{\textstyle{1 \over 2}}\]2}}\](1+e _T ), where the k=q–r _q–p basis elements e _T satisfy e _T ²=1. The lattice generated by a set of mutually annihilating primitive idempotents is examined. The final result characterizes all Clifford algebras R _{p, q} with an anti-involution such that each symmetric elements is either a nilpotent or then some right multiple of it is a nonzero symmetric idempotent. This happens when p+q<-3 and (p, q)(2, 1).     

Cyclic Relative Difference Sets and their p-Ranks

By modifying the constructions in Helleseth et al. [10] and No [15], we construct a family of cyclic ((q ^3k –1)/(q–1), q–1, q ^3k–1, q ^3k–2) relative difference sets, where q=3 ^e . These relative difference sets are liftings of the difference sets constructed in Helleseth et al. [10] and No [15]. In order to demonstrate that these relative difference sets are in general new, we compute p-ranks of the classical relative difference sets and 3-ranks of the newly constructed relative difference sets when q=3. By rank comparison, we show that the newly constructed relative difference sets are never equivalent to the classical relative difference sets, and are in general inequivalent to the affine GMW difference sets.     

Two-dimensional flag-transitive planes revisited

This paper shows that the odd order two-dimensional flag-transitive planes constructed by Kantor-Suetake constitute the same family of planes as those constructed by Baker-Ebert. Moreover, for orders satisfying a modest number theoretical assumption this family consists of all possible such planes of that order. In particular, it is shown that the number of isomorphism classes of (non-Desarguesian) two-dimensional flag-transitive affine planes of order q ² is precisely (q–1)/2 when q is an odd prime and precisely (q–1)/2e when q=p ^e is an odd prime power with exponent e that is a power of 2. An enumeration is given in other cases that uses the Möbius inversion formula.This work was partially supported by NSA grant MDA 904-95-H-1013.This work was partially supported by NSA grant MDA 904-94-H-2033.     

Orthogonal resolutions of lines in AG(n,q)

A resolution of the lines of AG(n,q) is a partition of the lines classes (called resolution classes) such that every point of the geometry is on exactly one line of each resolution class. Two resolutions R,R' of AG(n,q) are orthogonal if any resolution class from R has at most one line in common with any class from R'. In this paper, we construct orthogonal resolutions on AG(n,q) for all n=2ⁱ⁺¹, i=1,2,…, and all q>2 a prime power. The method involves constructing AG(n,q) from a finite projective plane of order q^n-1 and using the structure of the plane to display the orthogonal resolutions.     

On vector space partitions and uniformly resolvable designs

Let V _n (q) denote a vector space of dimension n over the field with q elements. A set of subspaces of V _n (q) is a partition of V _n (q) if every nonzero vector in V _n (q) is contained in exactly one subspace in . A uniformly resolvable design is a pairwise balanced design whose blocks can be resolved in such a way that all blocks in a given parallel class have the same size. A partition of V _n (q) containing a _i subspaces of dimension n _i for 1 ≤ i ≤ k induces a uniformly resolvable design on q ⁿ points with a _i parallel classes with block size , 1 ≤ i ≤ k, and also corresponds to a factorization of the complete graph into -factors, 1 ≤ i ≤ k. We present some sufficient and some necessary conditions for the existence of certain vector space partitions. For the partitions that are shown to exist, we give the corresponding uniformly resolvable designs. We also show that there exist uniformly resolvable designs on q ⁿ points where corresponding partitions of V _n (q) do not exist. A. D. Blinco—Part of this research was done while the author was visiting Illinois State University.     

Quasi-affine symmetric designs

A symmetric design with parameters v = q ²(q + 2), k = q(q + 1), λ = q, q ≥ 2, is called a quasi-affine design if its point set can be partitioned into q + 2 subsets P ₀, P ₁,..., P _q , P _q+1 such that the induced structure in every point neighborhood is an affine plane of order q (repeated q times). A quasi-affine design with q ≥ 3 determines its point neighborhoods uniquely and dual of such a design is also a quasi-affine design. These structural properties pave way for definition of a strongly quasi-affine design and it is also shown that associated with every quasi-affine design is a unique strongly quasi-affine design from which the given quasi-affine design is obtained by certain unique cutting and pasting operation. This investigation also enables us to associate a unique 2-regular graph with q + 2 vertices and in turn, a unique colored partition of the integer q + 2. These combinatorial consequences are finally used to obtain an exponential lower bound on the number of non-isomorphic solutions of such symmetric designs improving the earlier lower bound of 2. Work of Sanjeevani Gharge is supported by Faculty Improvement Programme of U.G.C., India.     

Correction to: “Exponential bounds on the number of designs with affine parameters”

It is well known that the number of designs with the parameters of a classical design having as blocks the hyperplanes in PG(n, q) or AG(n, q), n?3, grows exponentially. This result was extended recently [5] to designs having the same parameters as a projective geometry design whose blocks are the d‐subspaces of PG(n, q), for any 2?d?n ? 1. In this paper, exponential lower bounds are proved on the number of non‐isomorphic designs having the same parameters as an affine geometry design whose blocks are the d‐subspaces of AG(n, q), for any 2≤d≤n ? 1. Exponential bounds are also proved for the number of resolvable designs with these parameters. © 2011 Wiley Periodicals, Inc. J Combin Designs 19:156‐166, 2011     

Properties of convergence for -Bernstein polynomials

In this paper, we discuss properties of the ω,q-Bernstein polynomials introduced by S. Lewanowicz and P. Woźny in [S. Lewanowicz, P. Woźny, Generalized Bernstein polynomials, BIT 44 (1) (2004) 63–78], where fC[0,1], ω,q>0, ω≠1,q⁻¹,…,q⁻ⁿ⁺¹. When ω=0, we recover the q-Bernstein polynomials introduced by [G.M. Phillips, Bernstein polynomials based on the q-integers, Ann. Numer. Math. 4 (1997) 511–518]; when q=1, we recover the classical Bernstein polynomials. We compute the second moment of , and demonstrate that if f is convex and ω,q(0,1) or (1,∞), then are monotonically decreasing in n for all x[0,1]. We prove that for ω(0,1), q_n(0,1], the sequence converges to f uniformly on [0,1] for each fC[0,1] if and only if lim_n→∞q_n=1. For fixed ω,q(0,1), we prove that the sequence converges for each fC[0,1] and obtain the estimates for the rate of convergence of by the modulus of continuity of f, and the estimates are sharp in the sense of order for Lipschitz continuous functions.     

New Cyclic Difference Sets with Singer Parameters Constructed from d-Homogeneous Functions

In this paper, for a prime power q, new cyclic difference sets with Singer para- meters ((q ⁿ –1/q–1), (q ^n–1–1/q–1), (q ^n–2–1/q–1)) are constructed by using q-ary sequences (d-homogeneous functions) of period q ⁿ –1 and the generalization of GMW difference sets is proposed by combining the generation methods of d-form sequences and extended sequences. When q is a power of 3, new cyclic difference sets with Singer parameters ((q ⁿ –1/q–1), (q ^n–1–1/q–1), (q ^n–2–1/q–1)) are constructed from the ternary sequences of period q ⁿ –1 with ideal autocorrelation introduced by Helleseth, Kumar, and Martinsen.     

The Hering plane and associated designs

Absract We associate an affine planeA _ƒ with any automorphism ƒ of the additive group of the fieldF=GF(q), whereq is odd,F ^*=Δ ∪−Δ, and Δ=x ^ƒ x |x εF ^*. We compute the ternar of the planeA _ƒ. A simple construction of the Hering plane in the caseq=27,x ^ƒ=x−Trx=−x ³−x ⁹ and two designs associated with it are described in detail. Translated fromMatematicheskie Zametki, Vol. 60, No. 6, pp. 873–881, December, 1996.     

On tactical decompositions of class number 2

In this article we consider tactical decompositions of class number 2 of symmetric designs. Our main result says that if the orders are prime, then the only decompositions are of affine type. Moreover, we study symmetric decompositions of finite projective planes and show that, except in some cases, they are related to Baer subplanes, unitals, or 2 - ((m ² - m + 1)m, m, 1)designs.     

Konjugierte Netze mit p-ebenen u-Kurven,welche von den v-Kurven projektiv aufeinander bezogen werden

In this paper we consider conjugate nets in projective space P ⁿwith the following property: The u-curves of the net belong to p-dimensional subspaces of P ⁿand are projectively related to each other by the v-curves. We show that these nets form two classes. The first class consists of conjugate nets for which the u-curves from conic shadow boundaries. The u-curves of the nets of the second class, which we call C _p-nets, are rational normal curves of order p. Each u-curve possesses the characteristic (p–1)-space of its ambient p-space as a (p–1)-dimensional osculating space. This generalizes a result found by Degen [3] in the case p=2, n=3. By means of the Laplace transformation we get a construction of C _p-nets without integration.     

Recursive constructions for skew resolutions in affine geometries

A resolutionR inAG(n, q) is defined to be a partition of the lines into classesR ₁,R ₂, ...,R _t (t=(q ⁿ –1)/(q–1)) such that each point of the geometry is incident with precisely one line of each classR _l , 1it. Of course, the equivalence relation of parallelism defines a resolution in any affine geometry. A resolutionR is said to be a skew resolution provided noR _i , 1it, contains two parallel lines. Skew resolutions are useful for producing packings of lines in projective spaces and doubly resolvable block designs. Skew resolutions are known to exist inAG(n, q),n=2^t–1,i2,q a prime power. The entire spectrum is unknown. In this paper, we give two recursive constructions for skew resolutions. These constructions produce skew resolutions inAG(n, q) for infinietly many new values ofn.     

Building Symmetric Designs With Building Sets

We introduce a uniform technique for constructing a family of symmetric designs with parameters (v(q ^m+1-1)/(q-1), kq ^m ,q ^m), where m is any positive integer, (v, k, ) are parameters of an abelian difference set, and q = k ²/(k - ) is a prime power. We utilize the Davis and Jedwab approach to constructing difference sets to show that our construction works whenever (v, k, ) are parameters of a McFarland difference set or its complement, a Spence difference set or its complement, a Davis–Jedwab difference set or its complement, or a Hadamard difference set of order 9 · 4 ^d , thus obtaining seven infinite families of symmetric designs.