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In a linear code, a codeword is said to be minimal if its support does not contain the support of any other linearly independent codeword. Minimal codes endowed with the Hamming metric are linear codes whose codewords are all minimal. The study of these codes was motivated by their use for realizing perfect and ideal secret sharing schemes. In the first part of this talk we analyze the geometric point of view of minimal linear codes. Using the well-known correspondence between [n, k] codes over a finite field Fq and [n, k]q projective systems in PG(k − 1, q), it has been shown that equivalence classes of minimal linear codes are in one-toone correspondence with equivalence classes of cutting blocking sets. We give an overview of the bounds on the parameters of these codes and of the main constructions of small cutting blocking sets. In the second part of the talk we will focus on the geometric interpretation of rank-metric codes. In particular, we will consider the connection between rank-metric codes and the q-analogues of projective systems and apply it to minimal rank-metric codes. The correspondence between rank-metric codes and these geometric/combinatorial structures induces a correspondence between minimal rank-metric codes and linear cutting blocking sets. We will provide an overview of their existence results, parameters and constructions.

Joint work with Martino Borello, Alessandro Neri and Alberto Ravagnani