$t$-Fold $s$-Blocking Sets and $s$-Minimal Codes
Hao Chen, Xu Pan, Conghui Xie
TL;DR
The work strengthens the bridge between finite geometry and coding theory by deriving a stronger lower bound for t-fold s-blocking sets without t≤q, and by translating s-minimality into a cutting blocking-set framework. It generalizes the Ashikhmin-Barg criterion to s-minimal codes and delivers numerous infinite constructions, including Solomon-Stiffler-based families, while also presenting binary examples that are minimal but not 2-minimal. It further proves that (s+1)-minimal codes are automatically s-minimal and extends the theory with both constructive results and counterexamples to the AB condition. Collectively, the results yield new lower bounds on s-minimal code lengths and expand the toolkit for designing s-minimal codes with desired weight and blocking-set properties.
Abstract
Blocking sets and minimal codes have been studied for many years in projective geometry and coding theory. In this paper, we provide a new lower bound on the size of $t$-fold $s$-blocking sets without the condition $t \leq q$, which is stronger than the classical result of Beutelspacher in 1983. Then a lower bound on lengths of projective $s$-minimal codes is also obtained. It is proved that $(s+1)$-minimal codes are certainly $s$-minimal codes. We generalize the Ashikhmin-Barg condition for minimal codes to $s$-minimal codes. Many infinite families of $s$-minimal codes satisfying and violating this generalized Ashikhmin-Barg condition are constructed. We also give several examples which are binary minimal codes, but not $2$-minimal codes.