Steiner systems and configurations of points
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Steiner systems and configurations of points Edoardo Ballico1 · Giuseppe Favacchio2 · Elena Guardo2
· Lorenzo Milazzo2
Received: 17 September 2019 / Revised: 29 June 2020 / Accepted: 13 October 2020 © The Author(s) 2020
Abstract The aim of this paper is to make a connection between design theory and algebraic geometry/commutative algebra. In particular, given any Steiner System S(t, n, v) we associate two ideals, in a suitable polynomial ring, defining a Steiner configuration of points and its Complement. We focus on the latter, studying its homological invariants, such as Hilbert Function and Betti numbers. We also study symbolic and regular powers associated to the ideal defining a Complement of a Steiner configuration of points, finding its Waldschmidt constant, regularity, bounds on its resurgence and asymptotic resurgence. We also compute the parameters of linear codes associated to any Steiner configuration of points and its Complement. Keywords Steiner systems · Monomial ideals · Symbolic powers · Stanley Reisner rings · Linear codes Mathematics Subject Classification 51E10 · 13F55 · 13F20 · 14G50 · 94B27
1 Introduction Combinatorial design theory is the study of arranging elements of a finite set into patterns (subsets, words, arrays) according to specified rules. It is a field of combinatorics connected to several other areas of mathematics including number theory and finite geometries. In the
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Elena Guardo [email protected] http://www.dmi.unict.it/guardo Edoardo Ballico [email protected] Giuseppe Favacchio [email protected] https://sites.google.com/view/giuseppefavacchio Lorenzo Milazzo [email protected]
1
Dipartimento di Matematica, Via Sommarive, 14, 38123 Povo, TN, Italy
2
Dipartimento di Matematica e Informatica, Viale A. Doria, 6, 95100 Catania, Italy
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last years, the main techniques used in combinatorial and algebraic geometry allow design theory to grow up involving applications in other areas such as in coding theory, cryptography, and computer science. A t − (v, n, λ)-design D = (V , B) is a pair consisting of a set V of v points and a collection B of n-subsets of V , called blocks, such that every t-subset (or t-tuple) of V is contained in exactly λ blocks in B. The numbers v = |V |, b = |B|, n, λ, and t are called the parameters of the design. A Steiner system (V , B) of type S(t, n, v) is a t − (v, n, 1) design, that is, a collection B of n-subsets (blocks) of a v-set V such that each t-tuple of V is contained in a unique block in B. The elements in V are called vertices or points and those of B are called blocks. In particular, a Steiner triple system of order v, ST S(v), is a collection of triples (3-subsets) of V , such that each unordered pair of elements is contained in precisely one block, and a Steiner quadruple system of order v, S Q S(v), is a collection of quadruples (4-subsets) of V such that each triple is found in precisely one block. A geometric study of some particular classes of Steiner systems
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