A note on short minimal codes from subgeometries

2025/11/19 by Adriaensen, Sam, Sziklai, Peter, Weiner, Zsuzsa
#51E21 #94B27 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2511.15372

Abstract

In a 2022, Bartoli, Cossidente, Marino, and Pavese proved that in the projective space \rm PG(3,q³), 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(q²+1)(q+1),4]_q³ 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.

Citations

• Recent advances on minimal codes
• Strong blocking sets and minimal codes from expander graphs
• Short minimal codes and covering codes via strong blocking sets in\n projective spaces
• On cutting blocking sets and their codes
• Three Combinatorial Perspectives on Minimal Codes
• A geometric characterization of minimal codes and their asymptotic\n performance
• Full Characterization of Minimal Linear Codes as Cutting Blocking Sets
• Minimal linear codes arising from blocking sets
• Linear nonbinary covering codes and saturating sets in projective spaces

Related

• On cutting blocking sets and their codes
• Short minimal codes and covering codes via strong blocking sets in\n projective spaces
• Strong blocking sets and minimal codes from expander graphs
• Small Strong Blocking Sets by Concatenation
• Minimal codes from hypersurfaces in even characteristic