Full Characterization of Minimal Linear Codes as Cutting Blocking Sets

Abstract

In this paper, we first study more in detail the relationship between minimal linear codes and cutting blocking sets, which were recently introduced by Bonini and Borello, and then completely characterize minimal linear codes as cutting blocking sets. As a direct result, minimal projective codes of dimension 3 and t-fold blocking sets with t ≥ 2 in projective planes are identical objects. Some bounds on the parameters of minimal codes are derived from this characterization. Using this new link between minimal codes and blocking sets, we also present new general primary and secondary constructions of minimal linear codes. Resultantly, infinite families of minimal linear codes not satisfying the Aschikhmin-Barg's condition are obtained. In addition, the weight distributions of two subfamilies of the proposed minimal linear codes are established. Open problems are also presented.

A geometric characterization of minimal codes and their asymptotic performance

“…Finally, in all our constructions of minimal codes, and in other constructions provided in [8,9,12,36], we observed that the minimum distance satisfies d ≥ (k − 1)(q − 1) + 1, where this bound is met with equality in all our constructions of reduced minimal codes. Therefore, also motivated by Remark 4.4, where we observed that the bound of Theorem 4.3 is not sharp, we propose the following conjecture.…”

Three Combinatorial Perspectives on Minimal Codes

“…The previous bound is not tight in general. More precisely, in [1] it was conjectured (and then proved in [32]) that the length n of an [n, k] q minimal code satisfies the following lower bound.…”

“…The concept of a cutting blocking set was introduced in [13] for constructing a family of minimal codes. In [1] and [32] it was independently shown that cutting blocking sets are in one to one correspondence with minimal linear codes. In this subsection, we recall some properties of (cutting) blocking sets and known results about their size.…”

“…It is shown in [1,32] that the correspondence described above between projective [n, k, d] q systems and nondegenerate [n, k, d] q linear codes extends to a correspondence between equivalence classes of [n, k, d] q minimal codes and equivalence classes of projective [n, k, d] q systems that are cutting blocking sets. This geometric interpretation of minimal codes will be crucial in Section 4.…”

“…Constructions that exploit in other ways the minimal structure of the code are based, for example, on functions over finite fields [8,13,29,30]. A geometric approach was proposed in [1,27,32], where minimal codes are characterized as cutting blocking sets.…”