Steiner Triple Research Articles

Invertible Bloom Lookup Tables with Listing Guarantees

The Invertible Bloom Lookup Table (IBLT) is a probabilistic concise data structure for set representation that supports a listing operation as the recovery of the elements in the represented set. Its applications can be found in network synchronization and traffic monitoring as well as in error-correction codes. IBLT can list its elements with probability affected by the size of the allocated memory and the size of the represented set, such that it can fail with small probability even for relatively small sets. While previous works only studied the failure probability of IBLT, this work initiates the worst case analysis of IBLT that guarantees successful listing for all sets of a certain size. The worst case study is important since the failure of IBLT imposes high overhead. We describe a novel approach that guarantees successful listing when the set satisfies a tunable upper bound on its size. To allow that, we develop multiple constructions that are based on various coding techniques such as stopping sets and the stopping redundancy of error-correcting codes, and Steiner systems as well as new methodologies we develop. We analyze the sizes of IBLTs with listing guarantees obtained by the various methods as well as their mapping memory and runtime overheads. Lastly, we study lower bounds on the achievable sizes of IBLT with listing guarantees and verify the results in the paper by simulations.

On the (Non-)Existence of Tight Distance-Regular Graphs: a Local Approach

Let $\Gamma$ denote a distance-regular graph with diameter $D\geq 3.$ Jurišić and Vidali conjectured that if $\Gamma$ is tight with classical parameters $(D,b,\alpha,\beta)$, $b\geq 2$, then $\Gamma$ is not locally the block graph of an orthogonal array nor the block graph of a Steiner system. In the present paper, we prove this conjecture and, furthermore, extend it from the following aspect. Assume that for every triple of vertices $x, y, z$ of $\Gamma$, where $x$ and $y$ are adjacent, and $z$ is at distance $2$ from both $x$ and $y$, the number of common neighbors of $x$, $y$, $z$ is constant. We then show that if $\Gamma$ is locally the block graph of an orthogonal array (resp.~a Steiner system) with smallest eigenvalue $-m$, $m\geq 3$, then the intersection number $c_2$ is not equal to $m^2$ (resp. $m(m+1)$). Using this result, we prove that if a tight distance-regular graph $\Gamma$ is not locally the block graph of an orthogonal array or a Steiner system, then the valency (and hence diameter) of $\Gamma$ is bounded by a function in the parameter $b=b_1/(1+\theta_1)$, where $b_1$ is the intersection number of $\Gamma$ and $\theta_1$ is the second largest eigenvalue of $\Gamma$.

Conflict‐free hypergraph matchings

AbstractA celebrated theorem of Pippenger, and Frankl and Rödl states that every almost‐regular, uniform hypergraph with small maximum codegree has an almost‐perfect matching. We extend this result by obtaining a conflict‐free matching, where conflicts are encoded via a collection of subsets . We say that a matching is conflict‐free if does not contain an element of as a subset. Under natural assumptions on , we prove that has a conflict‐free, almost‐perfect matching. This has many applications, one of which yields new asymptotic results for so‐called ‘high‐girth’ Steiner systems. Our main tool is a random greedy algorithm which we call the ‘conflict‐free matching process’.

An explicit construction for large sets of infinite dimensional q $q$‐Steiner systems

AbstractLet be a vector space over the finite field . A ‐Steiner system, or an , is a collection of ‐dimensional subspaces of such that every ‐dimensional subspace of is contained in a unique element of . A large set of ‐Steiner systems, or an , is a partition of the ‐dimensional subspaces of into systems. In the case that has infinite dimension, the existence of an for all finite with was shown abstractly by Cameron in 1995. This paper provides an explicit construction of an for all prime powers , all positive integers , and where has countably infinite dimension.

Assignment of tasks to machines under data replication with a tie to Steiner systems

Assignment of tasks to machines under data replication with a tie to Steiner systems

Bussey systems and Steiner's tactical problem

In 1853, Steiner posed a number of combinatorial (tactical) problems, which eventually led to a large body of research on Steiner systems. However, solutions to Steiner's questions coincide with Steiner systems only for strengths two and three. For larger strengths, essentially only one class of solutions to Steiner's tactical problems is known, found by Bussey more than a century ago. In this paper, the relationships among Steiner systems, perfect binary one-error-correcting codes, and solutions to Steiner's tactical problem (Bussey systems) are discussed. For the latter, computational results are provided for at most 15 points.

Invertible Bloom Lookup Tables with Listing Guarantees

The Invertible Bloom Lookup Table (IBLT) is a probabilistic concise data structure for set representation that supports a listing operation as the recovery of the elements in the represented set. Its applications can be found in network synchronization and traffic monitoring as well as in error-correction codes. IBLT can list its elements with probability affected by the size of the allocated memory and the size of the represented set, such that it can fail with small probability even for relatively small sets. While previous works only studied the failure probability of IBLT, this work initiates the worst case analysis of IBLT that guarantees successful listing for all sets of a certain size. The worst case study is important since the failure of IBLT imposes high overhead. We describe a novel approach that guarantees successful listing when the set satisfies a tunable upper bound on its size. To allow that, we develop multiple constructions that are based on various coding techniques such as stopping sets and the stopping redundancy of error-correcting codes, and Steiner systems as well as new methodologies we develop. We analyze the sizes of IBLTs with listing guarantees obtained by the various methods as well as their mapping memory and runtime overheads. Lastly, we study lower bounds on the achievable sizes of IBLT with listing guarantees and verify the results in the paper by simulations.

Codes and pseudo-geometric designs from the ternary m-sequences with Welch-type decimation d = 2 ⋅ 3(n−1)/2 + 1

Pseudo-geometric designs are combinatorial designs which share the same parameters as a finite geometry design, but are not isomorphic to that design. As far as we know, many pseudo-geometric designs have been constructed by the methods of finite geometries and combinatorics. However, none of pseudo-geometric designs with the parameters S(2,q+1,(qn−1)/(q−1)) is constructed by the approach of coding theory. In this paper, we use cyclic codes to construct pseudo-geometric designs. We firstly present a family of ternary cyclic codes from the m-sequences with Welch-type decimation d=2⋅3(n−1)/2+1, and obtain some infinite family of 2-designs and a family of Steiner systems S(2,4,(3n−1)/2) using these cyclic codes and their duals. Moreover, the parameters of these cyclic codes and their shortened codes are also determined. Some of those ternary codes are optimal or almost optimal. Finally, we show that one of these obtained Steiner systems is inequivalent to the point-line design of the projective space PG(n−1,3) and thus is a pseudo-geometric design.

Mutually disjoint Steiner systems from BCH codes

Mutually disjoint Steiner systems from BCH codes

Ehrhart theory of paving and panhandle matroids

Abstract We show that the base polytope P M of any paving matroid M can be systematically obtained from a hypersimplex by slicing off certain subpolytopes, namely base polytopes of lattice path matroids corresponding to panhandle-shaped Ferrers diagrams. We calculate the Ehrhart polynomials of these matroids and consequently write down the Ehrhart polynomial of P M , starting with Katzman’s formula for the Ehrhart polynomial of a hypersimplex. The method builds on and generalizes Ferroni’s work on sparse paving matroids. Combinatorially, our construction corresponds to constructing a uniform matroid from a paving matroid by iterating the operation of stressed-hyperplane relaxation introduced by Ferroni, Nasr and Vecchi, which generalizes the standard matroid-theoretic notion of circuit-hyperplane relaxation. We present evidence that panhandle matroids are Ehrhart positive and describe a conjectured combinatorial formula involving chain forests and Eulerian numbers from which Ehrhart positivity of panhandle matroids will follow. As an application of the main result, we calculate the Ehrhart polynomials of matroids associated with Steiner systems and finite projective planes, and show that they depend only on their design-theoretic parameters: for example, while projective planes of the same order need not have isomorphic matroids, their base polytopes must be Ehrhart equivalent.

Towards a finer classification of strongly minimal sets

Let M be strongly minimal and constructed by a ‘Hrushovski construction’ with a single ternary relation. If the Hrushovski algebraization function μ is in a certain class T (μ triples) we show that for independent I with |I|>1, dcl⁎(I)=∅ (* means not in dcl of a proper subset). This implies the only definable truly n-ary functions f (f ‘depends’ on each argument), occur when n=1. We prove for Hrushovski's original construction and for the strongly minimal k-Steiner systems of Baldwin and Paolini that the symmetric definable closure, sdcl⁎(I)=∅ (Definition 2.7). Thus, no such theory admits elimination of imaginaries. As, we show that in an arbitrary strongly minimal theory, elimination of imaginaries implies sdcl⁎(I)≠∅. In particular, such strongly minimal Steiner systems with line-length at least 4 do not interpret a quasigroup, even though they admit a coordinatization if k=pn. The case structure depends on properties of the Hrushovski μ-function. The proofs depend on our introduction, for appropriate G⊆aut(M) (setwise or pointwise stabilizers of finite independent sets), the notion of a G-normal substructure A of M and of a G-decomposition of any finite such A. These results lead to a finer classification of strongly minimal structures with flat geometry, according to what sorts of definable functions they admit.

A family of diameter perfect constant-weight codes from Steiner systems

If S is a transitive metric space, then |C|⋅|A|≤|S| for any distance-d code C and a set A, “anticode”, of diameter less than d. For every Steiner S(t,k,n) system S, we show the existence of a q-ary constant-weight code C of length n, weight k (or n−k), and distance d=2k−t+1 (respectively, d=n−t+1) and an anticode A of diameter d−1 such that the pair (C,A) attains the code–anticode bound and the supports of the codewords of C are the blocks of S (respectively, the complements of the blocks of S). We study the problem of estimating the minimum value of q for which such a code exists, and find that minimum for small values of t.

The first infinite family of orthogonal Steiner systems S(3,5,v)

The research on orthogonal Steiner systems S(t,k,v) was initiated in 1968. For (t,k)∈{(2,3),(3,4)}, this corresponds to orthogonal Steiner triple systems (STSs) and Steiner quadruple systems (SQSs), respectively. The existence problem of a pair of orthogonal STSs or SQSs was settled completely thirty years ago. However, for Steiner systems with t≥3 and k≥5, only two small examples of orthogonal pairs were known to exist before this work. An infinite family of orthogonal Steiner systems S(3,5,v) is constructed in this paper. In particular, the existence of a pair of orthogonal Steiner systems S(3,5,4m+1) is established for any even m≥2; additionally a pair of orthogonal G-designs G(4m+15,5,5,3) is displayed for any odd m≥3. The construction is based on the Steiner systems admitting 3-transitive automorphism groups supported by elementary symmetric polynomials. Moreover, 50 mutually orthogonal Steiner systems S(5,8,24) are shown to exist.

Colorful Matchings

Suppose a committee consisting of three members is tasked with matching candidates to different positions. However, all of the committee members disagree on the job placement for every candidate, i.e., every candidate is matched to three different positions according to three committee members. All three committee members are competitive and want to push through as many of their placements as possible. Can they find a compromise which allows each committee member to be responsible for a third of all the candidate placements? In this paper we will consider an asymptotic version of this question and several other variants of a similar problem. As an application we will consider an embedding question—which hypertrees does a large Steiner system always contain?

Packings and Steiner systems in polar spaces

A finite classical polar space of rank \(n\) consists of the totally isotropic subspaces of a finite vector space equipped with a nondegenerate form such that \(n\) is the maximal dimension of such a subspace. A \(t\)-Steiner system in a finite classical polar space of rank \(n\) is a collection \(Y\) of totally isotropic \(n\)-spaces such that each totally isotropic \(t\)-space is contained in exactly one member of \(Y\). Nontrivial examples are known only for \(t=1\) and \(t=n-1\). We give an almost complete classification of such \(t\)-Steiner systems, showing that such objects can only exist in some corner cases. This classification result arises from a more general result on packings in polar spaces.Mathematics Subject Classifications: 51E23, 05E30, 33C80Keywords: Association schemes, codes, polar spaces, Steiner systems

On girth-biregular graphs

Let Γ denote a finite, connected, simple graph. For an edge e of Γ let n(e) denote the number of girth cycles containing e. For a vertex v of Γ let {e1, e2, …, ek} be the set of edges incident to v ordered such that n(e1) ≤ n(e2) ≤ … ≤ n(ek). Then (n(e1),n(e2),…,n(ek)) is called the signature of v. The graph Γ is said to be girth-biregular if it is bipartite, and all of its vertices belonging to the same bipartition have the same signature. Let Γ be a girth-biregular graph with girth g = 2d and signatures (a1,a2,…,ak1) and (b1,b2,…,bk2), and assume without loss of generality that k1 ≥ k2. Our first result is that {a1, a2, …, ak1} = {b1, b2, …, bk2}. Our next result is the upper bound ak1 ≤ M, where M = (k1−1)⌊g/4⌋(k2−1)⌈g/4⌉. We describe the graphs attaining equality. For d = 3 or d ≥ 4 even they are incidence graphs of Steiner systems and generalized polygons, respectively. Finally, we show that when d is even and ak1 = M − ε for some non-negative integer ε < k2 − 1, then ε = 0. Similar result is valid for d = 3, ε ≤ 1 and k2|̸ k1.

Latin Matchings and Ordered Designs OD(n−1, n, 2n−1)

This paper revisits a combinatorial structure called the large set of ordered design (LOD). Among others, we introduce a novel structure called Latin matching and prove that a Latin matching of order n leads to an LOD(n−1, n, 2n−1); thus, we obtain constructions for LOD(1, 2, 3), LOD(2, 3, 5), and LOD(4, 5, 9). Moreover, we show that constructing a Latin matching of order n is at least as hard as constructing a Steiner system S(n−2, n−1, 2n−2); therefore, the order of a Latin matching must be prime. We also show some applications in multiagent systems.

The prime submodules hypergraph of a free module of finite rank over a commutative ring

Let [Formula: see text] be a commutative ring with identity, [Formula: see text] be a prime ideal of [Formula: see text] and [Formula: see text] be a free [Formula: see text]-module of finite rank. In this paper, we use hypergraphs to give a full characterization of prime submodules of [Formula: see text]. Also we define a hypergraph [Formula: see text] called the prime submodules hypergraph of [Formula: see text] with respect to [Formula: see text] and use properties of Steiner systems for counting the number of [Formula: see text]-prime submodules of [Formula: see text] when the number of cosets of [Formula: see text] in [Formula: see text] is finite.

Block avoiding point sequencings of partial Steiner systems

A partial(n,k,t)_lambda -system is a pair (X,{mathcal {B}}) where X is an n-set of vertices and {mathcal {B}} is a collection of k-subsets of X called blocks such that each t-set of vertices is a subset of at most lambda blocks. A sequencing of such a system is a labelling of its vertices with distinct elements of {0,ldots ,n-1}. A sequencing is ell -block avoiding or, more briefly, ell -good if no block is contained in a set of ell vertices with consecutive labels. Here we give a short proof that, for fixed k, t and lambda , any partial (n,k,t)_lambda -system has an ell -good sequencing for some ell =Theta (n^{1/t}) as n becomes large. This improves on results of Blackburn and Etzion, and of Stinson and Veitch. Our result is perhaps of most interest in the case k=t+1 where results of Kostochka, Mubayi and Verstraëte show that the value of ell cannot be increased beyond Theta ((n log n)^{1/t}). A special case of our result shows that every partial Steiner triple system (partial (n,3,2)_1-system) has an ell -good sequencing for each positive integer ell leqslant 0.0908,n^{1/2}.

An infinite family of antiprimitive cyclic codes supporting Steiner systems $$S(3,8, 7^m+1)$$

Coding theory and combinatorial $t$-designs have close connections and interesting interplay. One of the major approaches to the construction of combinatorial t-designs is the employment of error-correcting codes. As we all known, some $t$-designs have been constructed with this approach by using certain linear codes in recent years. However, only a few infinite families of cyclic codes holding an infinite family of $3$-designs are reported in the literature. The objective of this paper is to study an infinite family of cyclic codes and determine their parameters. By the parameters of these codes and their dual, some infinite family of $3$-designs are presented and their parameters are also explicitly determined. In particular, the complements of the supports of the minimum weight codewords in the studied cyclic code form a Steiner system. Furthermore, we show that the infinite family of cyclic codes admit $3$-transitive automorphism groups.