Strong blocking sets and minimal codes from expander graphs

• DOI: 10.1090/tran/9205
• Publication Date: 2024-05-24T21:30:11Z
• Authors: Noga Alon; Anurag Bishnoi; Shagnik Das; Alessandro Neri
• Keywords: LINEAR CODES; FOS: Computer and information sciences; FUNCTION-FIELDS; Computer Science - Information Theory; Information Theory (cs.IT); 0102 computer and information sciences; EXPLICIT CONSTRUCTIONS; CURVES; 01 natural sciences; INTEGRITY; Mathematics and Statistics; FOS: Mathematics; Mathematics - Combinatorics; Combinatorics (math.CO)

A strong blocking set in a finite projective space is of points that intersects each hyperplane spanning set. We provide new graph theoretic construction such sets: combining constant-degree expanders with asymptotically good codes, we explicitly construct sets the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis k minus 1 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>k</mml:mi> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(k-1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-dimensional over alttext="double-struck upper F Subscript q"> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">F</mml:mi> <mml:mi>q</mml:mi> </mml:msub> encoding="application/x-tex">\mathbb {F}_q</mml:annotation> </inline-formula> have size at most alttext="c q k"> <mml:mi>c</mml:mi> encoding="application/x-tex">c k</mml:annotation> for some universal constant alttext="c"> encoding="application/x-tex">c</mml:annotation> </inline-formula>. Since recently been shown to be equivalent minimal linear our gives first explicit </inline-formula>-linear codes length alttext="n"> <mml:mi>n</mml:mi> encoding="application/x-tex">n</mml:annotation> and dimension alttext="k"> encoding="application/x-tex">k</mml:annotation> </inline-formula>, every prime power alttext="q"> encoding="application/x-tex">q</mml:annotation> which alttext="n less-than-or-equal-to c <mml:mo>≤</mml:mo> encoding="application/x-tex">n \leq This solves one main open problems on codes.