Full Characterization of Minimal Linear Codes as Cutting Blocking Sets
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by
Chunming Tang, Yan Qiu, Qunying Liao, Zhengchun Zhou
2019
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.
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