The application of invariant theory to the existence of quasi-symmetric designs

Gleason and Mallows and Sloane characterized the weight enumerators of maximal self-orthogonal codes with all weights divisible by 4. We apply these results to obtain a new necessary condition for the existence of 2 − (v, k, λ) designs where the intersection numbers s₁…,s_n satisfy s₁ ≡ s₂ ≡ … ≡ s_n (mod 2). Non-existence of quasi-symmetric 2−(21, 18, 14), 2−(21, 9, 12), and 2−(35, 7, 3) designs follows directly from the theorem. We also eliminate quasi-symmetric 2−(33, 9, 6) designs. We prove that the blocks of quasi-symmetric 2−(19, 9, 16), 2−(20, 10, 18), 2-(20,8, 14), and 2−(22, 8, 12) designs are obtained from octads and dodecads in the [24, 12] Golay code. Finally we eliminate quasi-symmetric 2−(19,9, 16) and 2-(22, 8, 12) designs.     

Characterizing geometric designs, II

We provide a characterization of the classical point-line designs PG₁(n,q), where n?3, among all non-symmetric 2-(v,k,1)-designs as those with the maximal number of hyperplanes. As an application of this result, we characterize the classical quasi-symmetric designs PGn−2(n,q), where n?4, among all (not necessarily quasi-symmetric) designs with the same parameters as those having line size q+1 and all intersection numbers at least qn−4+?+q+1. Finally, we also give an explicit lower bound for the number of non-isomorphic designs having the same parameters as PG₁(n,q); in particular, we obtain a new proof for the known fact that this number grows exponentially for any fixed value of q.     

Quasi-symmetric designs related to the triangular graph

LetT _m be the adjacency matrix of the triangular graph. We will give conditions for a linear combination ofT _m, I andJ to be decomposable. This leads to Bruck-Ryser-Chowla like conditions for, what we call, triangular designs. These are quasi-symmetric designs whose block graph is the complement of the triangular graph. For these designs our conditions turn out to be stronger than the known non-existence results for quasi-symmetric designs.     

Further characterizations of cubic lattice graphs

A cubic lattice graph with characteristic n is a graph whose points can be identified with the ordered triplets on n symbols and two points are adjacent whenever the corresponding triplets have two coordinates in common. An L₂ graph is a graph whose points can be identified with the ordered pairs on n symbols such that two points are adjacent if and only if the corresponding pairs have a common coordinate. The main result of this paper is two new characterizations and shows the relation between cubic lattice and L₂ graphs. The main result also suggests a conjecture concerning the characterization of interchange graphs of complete m-partite graphs.     

Inner balance of symmetric designs

A triple array is a row-column design which carries two balanced incomplete block designs (BIBDs) as substructures. McSorley et al. (Des Codes Cryptogr 35: 21–45, 2005), Section 8, gave one example of a triple array that also carries a third BIBD, formed by its row-column intersections. This triple array was said to be balanced for intersection, and they made a search for more such triple arrays among all potential parameter sets up to some limit. No more examples were found, but some candidates with suitable parameters were suggested. We define the notion of an inner design with respect to a block for a symmetric BIBD and present criteria for when this inner design can be balanced. As triple arrays in the canonical case correspond to SBIBDs, this in turn yields new existence criteria for triple arrays balanced for intersection. In particular, we prove that the residual design of the related SBIBD with respect to the defining block must be quasi-symmetric, and give necessary and sufficient conditions on the intersection numbers. This, together with our parameter bounds enable us to exclude the suggested triple array candidates in McSorley et al. (Des Codes Cryptogr 35: 21–45, 2005) and many others in a wide search. Further we investigate the existence of SBIBDs whose inner designs are balanced with respect to every block. We show as a key result that such SBIBDs must possess the quasi-3 property, and we answer the existence question for all known classes of these designs.     

Quasi-symmetric 2, 3, 4-designs

Quasi-symmetric designs are block designs with two block intersection numbersx andy It is shown that with the exception of (x, y)=(0, 1), for a fixed value of the block sizek, there are finitely many such designs. Some finiteness results on block graphs are derived. For a quasi-symmetric 3-design with positivex andy, the intersection numbers are shown to be roots of a quadratic whose coefficients are polynomial functions ofv, k and λ. Using this quadratic, various characterizations of the Witt—Lüneburg design on 23 points are obtained. It is shown that ifx=1, then a fixed value of λ determines at most finitely many such designs.     

Planar and affine spaces

In this note, we characterize finite three-dimensional affine spaces as the only linear spaces endowed with set Ω of proper subspaces having the properties (1) every line contains a constant number of points, say n, with n>2; (2) every triple of noncollinear points is contained in a unique member of Ω; (3) disjoint or coincide is an equivalence relation in Ω with the additional property that every equivalence class covers all points. We also take a look at the case n=2 (in which case we have a complete graph endowed with a set Ω of proper complete subgraphs) and classify these objects: besides the affine 3-space of order 2, two small additional examples turn up. Furthermore, we generalize our result in the case of dimension greater than three to obtain a characterization of all finite affine spaces of dimension at least 3 with lines of size at least 3.     

Recent results on designs with classical parameters

We report on recent results concerning designs with the same parameters as the classical geometric designs PG _d (n, q) formed by the points and d-dimensional subspaces of the n-dimensional projective space PG(n, q) over the field GF(q) with q elements, where 1 ???d ???n?1. The corresponding case of designs with the same parameters as the classical geometric designs AG _d (n, q) formed by the points and d-dimensional subspaces of the n-dimensional affine space AG(n, q) will also be discussed, albeit in less detail.     

Maximal invariant subspaces for a class of operators

In this note, we characterize maximal invariant subspaces for a class of operators. Let T be a Fredholm operator and \(1-TT^{*}\in\mathcal{S}_{p}\) for some p≥1. It is shown that if M is an invariant subspace for T such that dim?M ? TM<∞, then every maximal invariant subspace of M is of codimension 1 in M. As an immediate consequence, we obtain that if M is a shift invariant subspace of the Bergman space and dim?M ? zM<∞, then every maximal invariant subspace of M is of codimension 1 in M. We also apply the result to translation operators and their invariant subspaces.     

Contact points of convex bodies

LetB be a convex body in ? ⁿ and let ? be an ellipsoid of minimal volume containingB. By contact points ofB we mean the points of the intersection between the boundaries ofB and ?. By a result of P. Gruber, a generic convex body in ? ⁿ has (n+3)·n/2 contact points. We prove that for every ?>0 and for every convex bodyB ? ? ⁿ there exists a convex bodyK having $$m \leqslant C(\varepsilon ) \cdot n\log ^3 n$$ contact points whose Banach-Mazur distance toB is less than 1+?. We prove also that for everyt>1 there exists a convex symmetric body Γ ? ? ⁿ so that every convex bodyD ? ? ⁿ whose Banach-Mazur distance to Γ is less thant has at least (1+c ₀/t ²)·n contact points for some absolute constantc ₀. We apply these results to obtain new factorizations of Dvoretzky-Rogers type and to estimate the size of almost orthogonal submatrices of an orthogonal matrix.     

Spatial curve singularities and the monster/semple tower

The Monster tower ([MZ01], [MZ10]), known as the Semple Tower in Algebraic Geometry ([Sem54], [Ber10]), is a tower of fibrations canonically constructed over an initial smooth n-dimensional base manifold. Each consecutive fiber is a projective n — 1 space. Each level of the tower is endowed with a rank n distribution, that is, a subbundle of its tangent bundle. The pseudogroup of diffeomorphisms of the base acts on each level so as to preserve the fibration and the distribution. The main problem is to classify orbits (equivalence classes) relative to this action. Analytic curves in the base can be prolonged (= Nash blown-up) to curves in the tower which are integral for the distribution. Prolongation yields a dictionary between singularity classes of curves in the base n-space and orbits in the tower. This dictionary yielded a rather complete solution to the classification problem for n = 2 ([MZ10]). A key part of this solution was the construction of the ‘RVT’ classes, a discrete set of equivalence classes built from verifying conditions of transversality or tangency to the fiber at each level ([MZ10]). Here we define analogous ‘RC’ classes for n > 2 indexed by words in the two letters, R (for regular, or transverse) and C (for critical, or tangent). There are 2 ^k?1 such classes of length k and they exhaust the tower at level k. The codimension of such a class is the number of C’s in its word. We attack the classification problem by codimension, rather than level. The codimension 0 class is open and dense and its structure is well known. We prove that any point of any codimension 1 class is realized by a curve having a classical A _2k singularity (k depending on the type of class). Following ([MZ10]) we define what it means for a singularity class in the tower to be “tower simple”. The codimension 0 and 1 classes are tower simple, and tower simple implies simple in the usual sense of singularity. Our main result is a classification of the codimension 2 tower simple classes in any dimension n. A key step in the classification asserts that any point of any codimension 2 singularity is realized by a curve of multiplicity 3 or 4. A central tool used in the classification are the listings of curve singularities due to Arnol’d ([Arn99], Bruce-Gaffney ([BG82]), and Gibson-Hobbs ([GH93]). We also classify the first occurring truly spatial singularities as subclasses of the codimension 2 classes. (A point or a singularity class is “spatial” if there is no curve which realizes it and which can be made to lie in some smooth surface.) As a step in the classification theorem we establish the existence of a canonical arrangement of hyperplanes at each point, lying in the distribution n-plane at that point. This arrangement leads to a coding scheme finer than the RC coding. Using the arrangement coding we establish the lower bound of 29 for the number of distinct orbits in the case n = 3 and level 4. Finally, Mormul ([Mor04], [Mor09]) has defined a different coding scheme for singularity classes in the tower and in an appendix we establish some relations between our coding and his.     

Mixed partitions and related designs

We define a mixed partition of Π =  PG(d, q ^r ) to be a partition of the points of Π into subspaces of two distinct types; for instance, a partition of PG(2n ? 1, q ²) into (n ? 1)-spaces and Baer subspaces of dimension 2n ? 1. In this paper, we provide a group theoretic method for constructing a robust class of such partitions. It is known that a mixed partition of PG(2n ? 1, q ²) can be used to construct a (2n ? 1)-spread of PG(4n ? 1, q) and, hence, a translation plane of order q ²ⁿ . Here we show that our partitions can be used to construct generalized Andrè planes, thereby providing a geometric representation of an infinite family of generalized Andrè planes. The results are then extended to produce mixed partitions of PG(rn ? 1, q ^r ) for r ≥ 3, which lift to (rn ? 1)-spreads of PG(r ² n ? 1, q) and hence produce $2-(q^{r^2n},q^{rn},1)$ (translation) designs with parallelism. These designs are not isomorphic to the designs obtained from the points and lines of AG(r, q ^rn ).     

Codimension and pseudometric in co-Heyting algebras

In this paper, we introduce a notion of dimension and codimension for every element of a bounded distributive lattice L. These notions prove to have a good behavior when L is a co-Heyting algebra. In this case the codimension gives rise to a pseudometric on L which satisfies the ultrametric triangle inequality. We prove that the Hausdorff completion of L with respect to this pseudometric is precisely the projective limit of all its finite dimensional quotients. This completion has some familiar metric properties, such as the convergence of every monotonic sequence in a compact subset. It coincides with the profinite completion of L if and only if it is compact or equivalently if every finite dimensional quotient of L is finite. In this case we say that L is precompact. If L is precompact and Hausdorff, it inherits many of the remarkable properties of its completion, specially those regarding the join/meet irreducible elements. Since every finitely presented co-Heyting algebra is precompact Hausdorff, all the results we prove on the algebraic structure of the latter apply in particular to the former. As an application, we obtain the existence for every positive integers n, d of a term t _{n, d} such that in every co-Heyting algebra generated by an n-tuple a, t _{n, d} (a) is precisely the maximal element of codimension d.     

Subsets and subspaces

Several rules known to be valid within the set theory are revisited and considered from the point of view of the question whether they remain satisfied when sets are replaced with subspaces from C_n,1. It is shown that some of the set relationships hold true upon such a replacement, however, there is also an extensive class of relationships whose validity is no longer guaranteed; in this class the commutativity of the orthogonal projectors onto the involved subspaces proves to play an important role. Some related results inspired by probability theory are established as well.     

Banach spaces in which a theorem of orlicz is not true

Let the Banach space X be such that for every numerical sequencet _n ↘0 there exists in X an unconditionally convergent series σx_n, the terms of which are subject to the condition ∥x_n∥=t_n (n=1,2,...). Then $$\mathop {sup}\limits_n \mathop {inf}\limits_{X_n } d(X_n ,l_\infty ^n )< \infty ,$$ where X_n ranges over all the n-dimensional subspaces of X.     

Partitions of finite vector spaces into subspaces

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 element of V_n(q) is contained in exactly one element of . Suppose there exists a partition of V_n(q) into x_i subspaces of dimension n_i, 1 ≤ i ≤ k. Then x₁, …, x_k satisfy the Diophantine equation . However, not every solution of the Diophantine equation corresponds to a partition of V_n(q). In this article, we show that there exists a partition of V_n(2) into x subspaces of dimension 3 and y subspaces of dimension 2 if and only if 7x + 3y = 2ⁿ ? 1 and y ≠ 1. In doing so, we introduce techniques useful in constructing further partitions. We also show that partitions of V_n(q) induce uniformly resolvable designs on qⁿ points. © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 329–341, 2008     

Some characterizations of quasi-symmetric designs with a spread

The designPG ₂ (4,q) of the points and planes ofPG (4,q) forms a quasi-symmetric 2-design with block intersection numbersx=1 andy=q+1. We give some characterizations of quasi-symmetric designs withx=1 which have a spread through a fixed point. For instance, it is proved that if such a designD is also smooth, thenDPG ₂ (4,q).     

Admissible parameters of symmetric designs satisfying v=4(k-\lambda )+2 and symmetric designs with inner balance

The admissible parameters of symmetric \((v,k,\lambda )\) designs satisfying \(v=4(k-\lambda )+2\) are shown to correspond with the solutions of a certain Pell equation. We then determine the feasible parameters of such designs that could have a quasi-symmetric residual design with respect to a block, and classify them into two possible families. Finally, we consider the feasible parameters of symmetric designs with inner balance as defined by Nilson and Heidtmann (Des. Codes Cryptogr. doi:10.1007/s10623-012-9730-2, (2012)), and show that (with one exception) they must all belong to one of these families.     

Parameters for which the Griesmer bound is not sharp

We prove for a large class of parameters t and r that a multiset of at most tθ_d-k+rθ_d-k-2 points in PG(d,q) that blocks every k-dimensional subspace at least t times must contain a sum of t subspaces of codimension k.We use our results to identify a class of parameters for linear codes for which the Griesmer bound is not sharp. Our theorem generalizes the non-existence results from Maruta [On the achievement of the Griesmer bound, Des. Codes Cryptogr. 12 (1997) 83-87] and Klein [On codes meeting the Griesmer bound, Discrete Math. 274 (2004) 289-297].