Short Minimal Codes and Covering Codes via Strong Blocking Sets in Projective Spaces

Minimal linear codes are in one-to-one correspondence with special types of blocking sets projective spaces over a finite field, which called strong or cutting sets. have been studied since decades but their tight connection was unfolded only the past few years, and it has not fully exploited yet. In this paper we apply geometric probabilistic arguments to contribute field minimal codes. We prove an upper bound on length dimension $k$ notation="LaTeX">$q$ -element Galois is both , hence improve previous superlinear bounds. This result determines up small constant factor. also lower bounds size so higgledy-piggledy line these results present improved covering saturating as well.