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Minimal linear codes are in one-to-one correspondence with special types ofblocking sets of projective spaces over a finite field, which are called strongor cutting blocking sets. In this paper we prove an upper bound on the minimallength of minimal codes of dimension $k$ over the $q$-element Galois fieldwhich is linear in both $q$ and $k$, hence improve the previous superlinearbounds. This result determines the minimal length up to a small constantfactor. We also improve the lower and upper bounds on the size of so calledhiggledy-piggledy line sets in projective spaces and apply these results topresent improved bounds on the size of covering codes and saturating sets inprojective spaces as well. The contributions rely on geometric andprobabilistic arguments.

Short minimal codes and covering codes via strong blocking sets in projective spaces