Full Characterization of Minimal Linear Codes as Cutting Blocking Sets

In this paper, we first study in detail the relationship between minimal linear codes and cutting blocking sets, 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...

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Bibliographic Details
Published in:IEEE transactions on information theory 2021-06, Vol.67 (6), p.3690-3700
Main Authors: Tang, Chunming, Qiu, Yan, Liao, Qunying, Zhou, Zhengchun
Format: Article
Language:English
Subjects:
blocking set
Codecs
Codes
Cryptography
Cutting
Geometry
Hamming weight
hyperplane
Indexes
Linear code
Linear codes
minimal code
Parameters
secret sharing
Software
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Summary:In this paper, we first study in detail the relationship between minimal linear codes and cutting blocking sets, 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\ge 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. As a result, infinite families of minimal linear codes not satisfying the Aschikhmin-Barg's condition are obtained. In addition to this, open problems on the parameters and the weight distributions of some generated linear codes are presented.
ISSN:0018-9448
1557-9654
DOI:10.1109/TIT.2021.3070377