Full Characterization of Minimal Linear Codes as Cutting Blocking Sets

Linear Cutting Blocking Sets and Minimal Codes in the Rank Metric

“…Minimal rank-metric codes are the natural analogues (in the rank-metric) of minimal Hamming-metric codes, a class of objects that have been extensively studied in connection with finite geometry; see e.g. [2,10,43].…”

Small Strong Blocking Sets by Concatenation

“…In [10] they were reintroduced, with the name of cutting blocking sets, in order to construct a particular family of minimal codes. In [1] and [31] it was independently shown that strong blocking sets are the geometrical counterparts of minimal codes: the above correspondence between non-degenerate codes and projective systems restricts to a correspondence between equivalence classes of projective [n, k, d] q minimal codes and equivalence classes of subsets of PG(k − 1, q) that are strong blocking sets (since in this paper we are interested in short minimal codes or, equivalently, in small strong blocking sets in projective spaces, it is not restrictive to narrow down to projective codes, which correspond to projective systems in which all the points have multiplicity one, that is subsets). Clearly, the simplex codes defined in Example 1.2 correspond to the strong blocking sets defined in Example 1.4.…”

“…Strong blocking sets are the geometrical counterpart of an important class of error correcting codes, the so-called minimal codes; see [1,31]. A codeword is said to be minimal if its support does not contain the support of any other non-proportional nonzero codeword.…”

“…Bounds on the parameters of a minimal code were considered in [1,2,12,14,25]. Only recently, minimal codes were studied through their connection with strong blocking sets [1,10,25,31]. It is worth mentioning that minimal codes form a class of asymptotically good codes [1,14].…”

On the maximum number of minimal codewords

“…They have also found applications in cryptography: in secret sharing schemes [19] and in secure two-party computation [9]. The set of minimal codewords is only known for a few classes of codes (see [1,6,7,8,11,12,18,20,21,22]) and, in general, it is a very hard problem to determine this set. In this work, we consider the following question: what is the maximum number of minimal codewords of linear codes of a given length and dimension?…”