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Explicit constructions of optimal blocking sets and minimal codes

Anurag Bishnoi, István Tomon

Abstract

A strong $s$-blocking set in a projective space is a set of points that intersects each codimension-$s$ subspace in a spanning set of the subspace. We present an explicit construction of such sets in a $(k - 1)$-dimensional projective space over $\mathbb{F}_q$ of size $O_s(q^s k)$, which is optimal up to the constant factor depending on $s$. This also yields an optimal explicit construction of affine blocking sets in $\mathbb{F}_q^k$ with respect to codimension-$(s+1)$ affine subspaces, and of $s$-minimal codes. Our approach is motivated by a recent construction of Alon, Bishnoi, Das, and Neri of strong $1$-blocking sets, which uses expander graphs with a carefully chosen set of vectors as their vertex set. The main novelty of our work lies in constructing specific hypergraphs on top of these expander graphs, where tree-like configurations correspond to strong $s$-blocking sets. We also discuss some connections to size-Ramsey numbers of hypergraphs, which might be of independent interest.

Explicit constructions of optimal blocking sets and minimal codes

Abstract

A strong -blocking set in a projective space is a set of points that intersects each codimension- subspace in a spanning set of the subspace. We present an explicit construction of such sets in a -dimensional projective space over of size , which is optimal up to the constant factor depending on . This also yields an optimal explicit construction of affine blocking sets in with respect to codimension- affine subspaces, and of -minimal codes. Our approach is motivated by a recent construction of Alon, Bishnoi, Das, and Neri of strong -blocking sets, which uses expander graphs with a carefully chosen set of vectors as their vertex set. The main novelty of our work lies in constructing specific hypergraphs on top of these expander graphs, where tree-like configurations correspond to strong -blocking sets. We also discuss some connections to size-Ramsey numbers of hypergraphs, which might be of independent interest.

Paper Structure

This paper contains 12 sections, 15 theorems, 30 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Our results
  3. Background
  4. Expander graphs
  5. Coding theory
  6. Points in general position
  7. The hypergraph of linear combinations
  8. Constructions of strong blocking sets
  9. Codimension 2
  10. Explicit constructions via size-Ramsey numbers
  11. Optimal constructions for large primes
  12. Optimal constructions for small primes

Key Result

Lemma 1

Let $q>p$ be primes congruent to 1 modulo 4. Then there is an explicit construction of an $(n,p+1,2\sqrt{p})$-graph for some $n=\Theta(q^3)$.

Figures (4)

  • Figure 1: An illustration of a 3-tree-like subhypergraph $K$ we might find. The dashed edges are the edges of the tree $T$, while the black and gray edges (which have size 2 and 3, respectively) are the edges of $K$. The edges of $K$ in order are $\{1,6\}, \{2,4,6\}, \{3,5,6\}, \{4,9\}, \{5,6,9\},\{6,10\},$$\{7,10\}, \{8,9,10\},\{9,11\}, \{10,11,12\}$.
  • Figure 2: An illustration of a 3-uniform tight path of length 6.
  • Figure 3: An illustration for the proof of Theorem \ref{['thm:large_prime']}. An example of an $\ell=4$-tree-like hypergraph we might build on the union of the edges $f_1,\dots,f_5$.
  • Figure 4: An illustration for the proof of Theorem \ref{['thm:small_q']}.

Theorems & Definitions (35)

  • Definition 1: Expander graph
  • Lemma 1
  • Lemma 2: Expander mixing lemma
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Definition 2: Minimal codes
  • ...and 25 more