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, 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 $t\ge 2$ in projective planes are identical objects. Some bounds on the parameters of minimal codes are derived from this characterization. This confirms a recent conjecture by Alfarano, Borello and Neri in [a geometric characterization of minimal codes and their asymptotic performance, arXiv:1911.11738, 2019] about a lower bound of the minimum distance of a minimal code. 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, the weight distributions of two subfamilies of the proposed minimal linear codes are established. Open problems are also presented.

[1]  Aiden A. Bruen.  Polynomial Multiplicties over Finite Fields and Intersection Sets , 1992, J. Comb. Theory, Ser. A.

[2]  Tor Helleseth,et al.  Linear Codes With Two or Three Weights From Weakly Regular Bent Functions , 2015, IEEE Transactions on Information Theory.

[3]  Matteo Bonini,et al.  Minimal linear codes arising from blocking sets , 2019, ArXiv.

[4]  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.

[5]  Péter Sziklai,et al.  Lines in Higgledy-Piggledy Arrangement , 2014, Electron. J. Comb..

[6]  R. C. Bose,et al.  A characterization of flat spaces in a finite geometry and the uniqueness of the hamming and the MacDonald codes , 1966 .

[7]  Xia Li,et al.  Four classes of minimal binary linear codes with wmin/wmax<1/2 derived from Boolean functions , 2020, Des. Codes Cryptogr..

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

[9]  Simeon Ball,et al.  Multiple Blocking Sets and Arcs in Finite Planes , 1996 .

[10]  Alessandro Neri,et al.  A geometric characterization of minimal codes and their asymptotic performance , 2019, Adv. Math. Commun..

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

[12]  H. Niederreiter,et al.  Two-Weight Codes Punctured from Irreducible Cyclic Codes , 2008 .

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

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

[15]  ROBERT E. JAMISON,et al.  Covering Finite Fields with Cosets of Subspaces , 1977, J. Comb. Theory, Ser. A.

[16]  Longjiang Qu,et al.  Three Classes of Minimal Linear Codes Over the Finite Fields of Odd Characteristic , 2019, IEEE Transactions on Information Theory.

[17]  Sihem Mesnager,et al.  A new class of three-weight linear codes from weakly regular plateaued functions , 2017, ArXiv.

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

[19]  Stefano Marcugini,et al.  Linear nonbinary covering codes and saturating sets in projective spaces , 2009, Adv. Math. Commun..

[20]  A. Bruen.  Blocking Sets in Finite Projective Planes , 1971 .

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

[22]  Tamás Szőnyi,et al.  Lacunary Polynomials, Multiple Blocking Sets and Baer Subplanes , 1999 .

[23]  Xiwang Cao,et al.  The Parameters of Minimal Linear Codes , 2019, ArXiv.

[24]  O. Antoine,et al.  Theory of Error-correcting Codes , 2022 .

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

[26]  Simeon Ball,et al.  Multiple blocking sets in finite projective spaces and improvements to the Griesmer bound for linear codes , 2009, Des. Codes Cryptogr..

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

[28]  Can Xiang,et al.  It is indeed a fundamental construction of all linear codes , 2016, ArXiv.

[29]  A. Bruen,et al.  Baer subplanes and blocking sets , 1970 .

[30]  Daniele Bartoli,et al.  An inductive construction of minimal codes , 2019, ArXiv.

[31]  Sihem Mesnager,et al.  Linear codes with few weights from weakly regular bent functions based on a generic construction , 2016, Cryptography and Communications.

[32]  Sihem Mesnager,et al.  Minimal Linear Codes From Characteristic Functions , 2020, IEEE Transactions on Information Theory.

[33]  Cunsheng Ding,et al.  Cyclotomic Linear Codes of Order $3$ , 2007, IEEE Transactions on Information Theory.

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