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A note on short minimal codes from subgeometries

Sam Adriaensen, Peter Sziklai, Zsuzsa Weiner

TL;DR

The paper addresses the existence of three $\mathbb{F}_q$-subgeometries in $\mathrm{PG}(3,q^3)$ whose union forms a strong blocking set, which yields linear minimal codes with parameters $[3(q^2+1)(q+1),4]_{q^3}$ for all prime powers $q$. It provides a short, geometrically flavored proof for odd $q>9$ by leveraging the theory of small blocking sets in projective planes and a construction based on disjoint subgeometries of subspaces. The key contribution is showing, via $\mathbb{F}_q$-subgeometries arranged through $R$-independence, that such unions yield strong blocking sets of the desired size, and applying this to reproduce the BCMP result with a streamlined argument. This work clarifies the connection between small blocking sets in $\mathrm{PG}(2,q)$ and minimal codes, offering a more accessible route to the known $[3(q^2+1)(q+1),4]_{q^3}$ codes and suggesting avenues for extensions to higher values of $k$.

Abstract

In a 2022, Bartoli, Cossidente, Marino, and Pavese proved that in the projective space ${\rm PG}(3,q^3)$, one can find three $\mathbb F_q$-subgeometries such that the union of their point sets is a strong blocking set. This proves the existence of linear minimal codes with parameters $[3(q^2+1)(q+1),4]_{q^3}$ for every prime power $q$. We give a short proof of this result for odd values of $q > 9$, using the theory of small blocking sets in projective planes.

A note on short minimal codes from subgeometries

TL;DR

The paper addresses the existence of three -subgeometries in whose union forms a strong blocking set, which yields linear minimal codes with parameters for all prime powers . It provides a short, geometrically flavored proof for odd by leveraging the theory of small blocking sets in projective planes and a construction based on disjoint subgeometries of subspaces. The key contribution is showing, via -subgeometries arranged through -independence, that such unions yield strong blocking sets of the desired size, and applying this to reproduce the BCMP result with a streamlined argument. This work clarifies the connection between small blocking sets in and minimal codes, offering a more accessible route to the known codes and suggesting avenues for extensions to higher values of .

Abstract

In a 2022, Bartoli, Cossidente, Marino, and Pavese proved that in the projective space , one can find three -subgeometries such that the union of their point sets is a strong blocking set. This proves the existence of linear minimal codes with parameters for every prime power . We give a short proof of this result for odd values of , using the theory of small blocking sets in projective planes.

Paper Structure

This paper contains 6 sections, 7 theorems, 11 equations.

Table of Contents

  1. Keywords.
  2. MSC.
  3. Introduction
  4. Strong blocking sets from disjoint subgeometries
  5. Reproving Result
  6. Acknowledgements.

Key Result

Lemma 6

Let $\alpha_1, \dots, \alpha_{k-1}$ be elements of $\mathbb F_{q^{k(k-1)}}$. Consider the set $S$ of points in $\mathop{\mathrm{PG}}\nolimits(k-1,q^{k-1})$ having a coordinate vector in $\bigcup_{i=1}^{k-1} \alpha_i R^*$. If $\alpha_1, \dots, \alpha_{k-1}$ are $R$-independent, then $S$ is a strong b

Theorems & Definitions (19)

  • Definition 1
  • Definition 3
  • Definition 5
  • Lemma 6
  • proof
  • Definition 7
  • Proposition 8
  • proof
  • Lemma 9
  • proof
  • ...and 9 more