Multiple blocking sets and multisets in Desarguesian planes

In AG(2, q ²), the minimum size of a minimal (q ? 1)-fold blocking set is known to be q ³ ? 1. Here, we construct minimal (q ? 1)-fold blocking sets of size q ³ in AG(2, q ²). As a byproduct, we also obtain new two-character multisets in PG(2, q ²). The essential idea in this paper is to investigate q ³-sets satisfying the opposite of Ebert’s discriminant condition.     

Hall sets,Lazard sets and comma-free codes

We investigate the relationship between two constructions of maximal comma-free codes described, respectively, by Eastman and by Scholtz and the notions of Hall sets and Lazard sets introduced in connection with factorizations of free monoids and bases of free Lie algebras.     

Coprimitive sets and inextendable codes

Complete (n,r)-arcs in PG(k−1,q) and projective (n,k,n−r) _q -codes that admit no projective extensions are equivalent objects. We show that projective codes of reasonable length admit only projective extensions. Thus, we are able to prove the maximality of many known linear codes. At the same time our results sharply limit the possibilities for constructing long non-linear codes. We also show that certain short linear codes are maximal. The methods here may be just as interesting as the results. They are based on the Bruen–Silverman model of linear codes (see Alderson TL (2002) PhD. Thesis, University of Western Ontario; Alderson TL (to appear) J Combin Theory Ser A; Bruen AA, Silverman R (1988) Geom Dedicata 28(1): 31–43; Silverman R (1960) Can J Math 12: 158–176) as well as the theory of Rédei blocking sets first introduced in Bruen AA, Levinger B (1973) Can J Math 25: 1060–1065.      

Intersection patterns of convex sets

LetK ₁,…K_n be convex sets inR ^d. For 0≦i denote byf _ithe number of subsetsS of {1,2,…,n} of cardinalityi+1 that satisfy ∩{K _i∶i∈S}≠Ø. We prove:Theorem.If f _d+r=0 for somer r>=0, then {fx161-1} This inequality was conjectured by Katchalski and Perles. Equality holds, e.g., ifK ₁=…=K_r=R^d andK _r+1,…,K_n aren?r hyperplanes in general position inR ^d. The proof uses multilinear techniques (exterior algebra). Applications to convexity and to extremal set theory are given.     

Intersection of independent regenerative sets

We investigate the nature of the intersection of two independent regenerative sets. The approach combines Bochners subordination and potential theory for a pair of Markov processes in duality. Received: 21 November 1997 / Revised version: 31 August 1998     

Intersection properties of finite sets

Let X₁, X₂, …, X_m be finite sets. The present paper is concerned with the m² ? m intersection numbers |X_i ∩ X_j| (i ≠ j). We prove several theorems on families of sets with the same prescribed intersection numbers. We state here one of our conclusions that requires no further terminology. Let T₁, T₂, …, T_m be finite sets and let m ? 3. We assume that each of the elements in the set union T₁ ∪ T₂ ∪ … ∪ T_m occurs in at least two of the subsets T₁, T₂, …, T_m. We further assume that every pair of sets T_i and T_j (i ≠ j) intersect in at most one element and that for every such pair of sets there exists exactly one set T_k (k ≠ i, k ≠ j) such that T_k intersects both T_i and T_j. Then it follows that the integer m = 2m′ + 1 is odd and apart from the labeling of sets and elements there exist exactly m′ + 1 such families of sets. The unique family with the minimal number of elements is {1}, {2}, …, {m′}, {1}, {2}, …, {m′}, {1, 2, …, m′}.     

Invariant sets under permutation,extremal graphs,and covering codes

Let $c_q(n,R)$ denote the minimum cardinality of a subset H in ${\mathbb{F}}_q^n$ such that every word in this space differs in at most R coordinates from a multiple of a vector in H, where q is a prime or a prime power. In order to explore the symmetries of such coverings, we investigate a few algebraic properties of invariant sets under permutation. Extremal problems arising from invariant sets are also studied on a graph theoretical viewpoint. As an application, a new class of upper bounds on $c_q(n,R)$ is constructed.     

Intersection queries in sets of disks

In this paper we develop some new data structures for storing a set of disks that can answer different types of intersection queries efficiency. If the disks are non-intersecting we obtain a linear size data structure that can report allk disks intersecting a query line segment in timeO(n ⁺ +k), wheren is the number of disks,=log₂(1+5)–1 0.695, and is an arbitrarily small positive constant. If the segment is a full line, the query time becomesO(n +k). For intersecting disks we obtain anO(n logn) size data structure that can answer an intersection query in timeO(n ^2/3 log² n+k). We also present a linear size data structure for ray shooting queries, whose query time isO(n ).The research of the first two authors was supported by the ESPRIT Basic Research Action No. 3075 (project ALCOM). The work of the third author was supported byDimacs (Center for Discrete Mathematics and Theoretical Computer Science), a National Science Foundation Science and Technology Center — NSF-STC88-09648.     

Affine blocking sets, three-dimensional codes and the Griesmer bound

We construct families of three-dimensional linear codes that attain the Griesmer bound and give a non-explicit construction of linear codes that are one away from the Griesmer bound. All these codes contain the all-1 codeword and are constructed from small multiple blocking sets in AG(2,q).     

Intersection of Hamming codes avoiding Hamming subcodes

We prove that given a binary Hamming code \({{\mathcal{H}}^n}\) of length n = 2 ^m ? 1, m ≥ 3, or equivalently a projective geometry PG(m ? 1, 2), there exist permutations \({\pi \in \mathcal{S}_n}\) , such that \({{\mathcal{H}}^n}\) and \({\pi({\mathcal{H}}^n)}\) do not have any Hamming subcode with the same support, or equivalently the corresponding projective geometries do not have any common flat. The introduced permutations are called AF permutations. We study some properties of these permutations and their relation with the well known APN functions.     

Information sets in abelian codes: defining sets and Groebner basis

In Bernal and Simón (IEEE Trans Inf Theory 57(12):7990–7999, 2011) we introduced a technique to construct information sets for every semisimple abelian code by means of its defining set. This construction is a non trivial generalization of that given by Imai (Inf Control 34:1–21, 1977) in the case of binary two-dimensional cyclic (TDC) codes. On the other hand, Sakata (IEEE Trans Inf Theory IT-27(5):556–565, 1981) showed a method for constructing information sets for binary TDC codes based on the computation of Groebner basis which agrees with the information set obtained by Imai. Later, Chabanne (IEEE Trans Inf Theory 38(6):1826–1829, 1992) presents a generalization of the permutation decoding algorithm for binary abelian codes by using Groebner basis, and as a part of his method he constructs an information set following the same ideas introduced by Sakata. In this paper we show that, in the general case of q-ary multidimensional abelian codes, both methods, that based on Groebner basis and that defined in terms of the defining sets, also yield the same information set.     

Multi-orbit cyclic subspace codes and linear sets

Cyclic subspace codes gained a lot of attention especially because they may be used in random network coding for correction of errors and erasures. Roth, Raviv and Tamo in 2018 established a connection between cyclic subspace codes (with certain parameters) and Sidon spaces. These latter objects were introduced by Bachoc, Serra and Zémor in 2017 in relation with the linear analogue of Vosper's Theorem. This connection allowed Roth, Raviv and Tamo to construct large classes of cyclic subspace codes with one or more orbits. In this paper we will investigate cyclic subspace codes associated to a set of Sidon spaces, that is cyclic subspace codes with more than one orbit. Moreover, we will also use the geometry of linear sets to provide some bounds on the parameters of a cyclic subspace code. Conversely, cyclic subspace codes are used to construct families of linear sets which extend a class of linear sets recently introduced by Napolitano, Santonastaso, Polverino and the author. This yields large classes of linear sets with a special pattern of intersection with the hyperplanes, defining rank metric and Hamming metric codes with only three distinct weights.     

Intersection theorems for group divisible difference sets

Let D be an (m,n;k;λ₁,λ₂)-group divisible difference set (GDDS) of a group G, written additively, relative to H, i.e. D is a k-element subset of G, H is a normal subgroup of G of index m and order n and for every nonzero element g of G,?{(d₁,d₂)?,d₁,d₂?D,d₁?d₂=g}? is equal to λ₁ if g is in H, and equal to λ₂ if g is not in H. Let H₁,H₂,…,H_m be distinct cosets of H in G and S_i=D∩H_i for all i=1,2,…,m. Some properties of S₁,S₂,…,S_m are studied here. Table 1 shows all possible cardinalities of S_i's when the order of G is not greater than 50 and not a prime. A matrix characterization of cyclic GDDS's with λ₁=0 implies that there exists a cyclic affine plane of even order, say n, only if n is divisible by 4 and there exists a cyclic (n?1,$\text{1}\text{2}$n?1,$\text{1}\text{4}$n?1)-difference set.