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Minimal Linear Codes Constructed from Sunflowers.

Xia Wu1, Wei Lu1

  • 1School of Mathematics, Southeast University, Nanjing 210096, China.

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|December 23, 2023
PubMed
Summary
This summary is machine-generated.

Sunflower codes are crucial subspace codes in coding theory. This study determines when linear codes derived from sunflowers are minimal, finding minimality depends on the sunflower size (s) relative to the finite field size (q).

Keywords:
linear codeminimal codesunflower

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Area of Science:

  • Coding Theory
  • Finite Fields
  • Algebraic Geometry

Background:

  • Sunflowers represent a significant class of subspace codes.
  • These codes are instrumental in the construction of linear codes.
  • The study of subspace code minimality is essential for understanding their properties and applications.

Purpose of the Study:

  • To investigate the minimality of linear codes constructed from sunflowers over finite fields (Fq).
  • To establish conditions on the sunflower size (s) and field size (q) that determine code minimality.

Main Methods:

  • Analysis of linear codes derived from sunflower structures.
  • Case-by-case examination of code minimality based on the relationship between s and q.

Main Results:

  • Linear codes from sunflowers are minimal when s ≥ q+1.
  • Codes are not minimal when 2 ≤ s ≤ 3 ≤ q.
  • For 3 < s ≤ q, minimality is dependent on the specific sunflower structure.

Conclusions:

  • The minimality of sunflower-based linear codes is contingent upon the interplay between sunflower size and finite field characteristics.
  • Clear thresholds for minimality are identified, with a nuanced outcome in intermediate cases (3 < s ≤ q).