Minimal linear codes arising from blocking sets

Minimal linear codes are algebraic objects which gained interest in the last 20 years, due to their link with Massey’s secret sharing schemes. In this context, Ashikhmin and Barg provided a useful and a quite easy-to-handle sufficient condition for a linear code to be minimal, which has been applied in the construction of many minimal linear codes. In this paper, we generalize some recent constructions of minimal linear codes which are not based on Ashikhmin–Barg’s condition. More combinatorial and geometric methods are involved in our proofs. In particular, we present a family of codes arising from particular blocking sets, which are well-studied combinatorial objects. In this context, we will need to define cutting blocking sets and to prove some of their relations with other notions in blocking sets’ theory. At the end of the paper, we provide one explicit family of cutting blocking sets and related minimal linear codes, showing that infinitely many of its members do not satisfy the Ashikhmin–Barg’s condition.

[1]  Seunghwan Chang,et al.  Linear codes from simplicial complexes , 2018, Des. Codes Cryptogr..

[2]  Cunsheng Ding,et al.  A Class of Two-Weight and Three-Weight Codes and Their Applications in Secret Sharing , 2015, IEEE Transactions on Information Theory.

[3]  Elwyn R. Berlekamp,et al.  On the inherent intractability of certain coding problems (Corresp.) , 1978, IEEE Trans. Inf. Theory.

[4]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[5]  Cunsheng Ding,et al.  Covering and Secret Sharing with Linear Codes , 2003, DMTCS.

[6]  J. Hirschfeld.  Projective Geometries Over Finite Fields , 1980 .

[7]  Sihem Mesnager,et al.  Linear codes from weakly regular plateaued functions and their secret sharing schemes , 2018, Des. Codes Cryptogr..

[8]  Alexander Barg,et al.  Minimal Vectors in Linear Codes , 1998, IEEE Trans. Inf. Theory.

[9]  JM Jeroen Doumen,et al.  Some applications of coding theory in cryptography , 2003 .

[10]  Cunsheng Ding,et al.  Minimal Linear Codes over Finite Fields , 2018, Finite Fields Their Appl..

[11]  W. Cary Huffman,et al.  Fundamentals of Error-Correcting Codes , 1975 .

[12]  Haode Yan,et al.  Four families of minimal binary linear codes with $$w_{\min }/w_{\max }\le 1/2$$wmin/wmax≤1/2 , 2018, Applicable Algebra in Engineering, Communication and Computing.

[13]  James L. Massey,et al.  Minimal Codewords and Secret Sharing , 1999 .

[14]  Gérard D. Cohen,et al.  Towards Secure Two-Party Computation from the Wire-Tap Channel , 2013, ICISC.

[15]  Moni Naor,et al.  The hardness of decoding linear codes with preprocessing , 1990, IEEE Trans. Inf. Theory.

[16]  Daniele Bartoli,et al.  Minimal Linear Codes in Odd Characteristic , 2019, IEEE Transactions on Information Theory.

[17]  Cunsheng Ding,et al.  Minimal Binary Linear Codes , 2018, IEEE Transactions on Information Theory.

[18]  L. Storme,et al.  Current Research Topics on Galois Geometry , 2011 .