14 research outputs found
A geometric characterization of minimal codes and their asymptotic performance
In this paper, we give a geometric characterization of minimal linear codes.
In particular, we relate minimal linear codes to cutting blocking sets,
introduced in a recent paper by Bonini and Borello. Using this
characterization, we derive some bounds on the length and the distance of
minimal codes, according to their dimension and the underlying field size.
Furthermore, we show that the family of minimal codes is asymptotically good.
Finally, we provide some geometrical constructions of minimal codes.Comment: 22 page
Strong blocking sets and minimal codes from expander graphs
A strong blocking set in a finite projective space is a set of points that
intersects each hyperplane in a spanning set. We provide a new graph theoretic
construction of such sets: combining constant-degree expanders with
asymptotically good codes, we explicitly construct strong blocking sets in the
-dimensional projective space over that have size . Since strong blocking sets have recently been shown to be equivalent to
minimal linear codes, our construction gives the first explicit construction of
-linear minimal codes of length and dimension , for every
prime power , for which . This solves one of the main open
problems on minimal codes.Comment: 20 page