Extending two families of maximum rank distance codes

In this paper we provide a large family of rank-metric codes, which contains properly the codes recently found by Longobardi and Zanella (2021) and by Longobardi, Marino, Trombetti and Zhou (2021). These codes are Fq2t -linear of dimension 2 in the space of linearized polynomials over Fq2t , where t is any integer greater than 2, and we prove that they are maximum rank distance codes. For t ≥ 5, we determine their equivalence classes and these codes turn out to be inequivalent to any other construction known so far, and hence they are really new.

[1]  Rod Gow,et al.  GALOIS EXTENSIONS AND SUBSPACES OF ALTERNATING BILINEAR FORMS WITH SPECIAL RANK PROPERTIES , 2009 .

[2]  Daniel Augot,et al.  Rank-metric codes over arbitrary Galois extensions and rank analogues of Reed-Muller codes , 2020, ArXiv.

[3]  E. Byrne,et al.  Tensor Representation of Rank-Metric Codes , 2019, SIAM J. Appl. Algebra Geom..

[4]  Katherine Morrison,et al.  Equivalence for Rank-Metric and Matrix Codes and Automorphism Groups of Gabidulin Codes , 2013, IEEE Transactions on Information Theory.

[5]  Giuseppe Marino,et al.  MRD-codes arising from the trinomial x q +x q 3 +cx q 5 ⊕F q 6 [x]. , 2019 .

[6]  Frank R. Kschischang,et al.  A Rank-Metric Approach to Error Control in Random Network Coding , 2007, IEEE Transactions on Information Theory.

[7]  Rocco Trombetti,et al.  On kernels and nuclei of rank metric codes , 2016, ArXiv.

[8]  Yue Zhou,et al.  MRD codes with maximum idealizers , 2020, Discret. Math..

[9]  Olga Polverino,et al.  Connections between scattered linear sets and MRD-codes , 2020, Bull. ICA.

[10]  Olga Polverino,et al.  Linear sets in finite projective spaces , 2010, Discret. Math..

[11]  Alberto Ravagnani,et al.  Rank-metric codes and q-polymatroids , 2018, Journal of Algebraic Combinatorics.

[12]  Gabriele Nebe,et al.  Automorphism groups of Gabidulin-like codes , 2016, ArXiv.

[13]  Raphael Overbeck,et al.  Structural Attacks for Public Key Cryptosystems based on Gabidulin Codes , 2008, Journal of Cryptology.

[14]  Giuseppe Marino,et al.  New maximum scattered linear sets of the projective line , 2017, Finite Fields Their Appl..

[15]  Daniel Augot,et al.  Rank metric and Gabidulin codes in characteristic zero , 2013, 2013 IEEE International Symposium on Information Theory.

[16]  Rocco Trombetti,et al.  Generalized Twisted Gabidulin Codes , 2015, J. Comb. Theory, Ser. A.

[17]  Ferdinando Zullo,et al.  Vertex properties of maximum scattered linear sets of PG(1, qn) , 2019, Discret. Math..

[18]  Alessandro Neri.  Algebraic theory of rank-metric codes : representations, invariants and density results , 2019 .

[19]  G. Longobardi,et al.  Linear sets and MRD-codes arising from a class of scattered linearized polynomials , 2021, Journal of Algebraic Combinatorics.

[20]  S. Chase,et al.  Galois Theory and Cohomology of Commutative Rings , 1978 .

[21]  John Sheekey,et al.  A characterization of the number of roots of linearized and projective polynomials in the field of coefficients , 2018, Finite Fields Their Appl..

[22]  Giuseppe Marino,et al.  Maximum scattered linear sets and MRD-codes , 2017, 1701.06831.

[23]  Guglielmo Lunardon,et al.  MRD-codes and linear sets , 2017, J. Comb. Theory, Ser. A.

[24]  Ernst M. Gabidulin,et al.  Ideals over a Non-Commutative Ring and thier Applications in Cryptology , 1991, EUROCRYPT.

[25]  Alessandro Neri,et al.  Systematic encoders for generalized Gabidulin codes and the q-analogue of Cauchy matrices , 2018, Linear Algebra and its Applications.

[26]  Philippe Delsarte,et al.  Bilinear Forms over a Finite Field, with Applications to Coding Theory , 1978, J. Comb. Theory A.

[27]  Sven Puchinger,et al.  Construction and decoding of evaluation codes in hamming and rank metric , 2018 .

[28]  Baofeng Wu,et al.  Linearized polynomials over finite fields revisited , 2012, Finite Fields Their Appl..

[29]  Michel Lavrauw,et al.  Field reduction and linear sets in finite geometry , 2013, 1310.8522.

[30]  Ferdinando Zullo,et al.  Scattered subspaces and related codes , 2020, Designs, Codes and Cryptography.

[31]  John Sheekey,et al.  A new family of linear maximum rank distance codes , 2015, Adv. Math. Commun..

[32]  Giuseppe Marino,et al.  A characterization of linearized polynomials with maximum kernel , 2018, Finite Fields Their Appl..

[33]  Ron M. Roth,et al.  Author's Reply to Comments on 'Maximum-rank array codes and their application to crisscross error correction' , 1991, IEEE Trans. Inf. Theory.

[34]  Marvin Anas Hahn,et al.  Valued rank-metric codes , 2021, ArXiv.

[35]  Giuseppe Marino,et al.  MRD-codes arising from the trinomial xq+xq3+cxq5⊕Fq6[x] , 2020, Linear Algebra and its Applications.

[36]  Giuseppe Marino,et al.  A large family of maximum scattered linear sets of $\mathrm{PG}(1,q^n)$ and their associated MRD codes , 2021 .

[37]  Christophe Mouaha,et al.  Rank-Metric Codes Over Finite Principal Ideal Rings and Applications , 2019, IEEE Transactions on Information Theory.

[38]  Alessandro Neri,et al.  Invariants and Inequivalence of Linear Rank-Metric Codes , 2019, 2019 IEEE International Symposium on Information Theory (ISIT).

[39]  Daniele Bartoli,et al.  A new family of maximum scattered linear sets in PG(1, q6) , 2020, Ars Math. Contemp..

[40]  Giuseppe Marino,et al.  A new family of MRD-codes , 2017, Linear Algebra and its Applications.

[41]  G. McGuire,et al.  Some results on linearized trinomials that split completely , 2019, Finite Fields and their Applications.

[42]  Ferdinando Zullo,et al.  Identifiers for MRD-codes , 2018, Linear Algebra and its Applications.

[43]  Anna-Lena Horlemann-Trautmann,et al.  New criteria for MRD and Gabidulin codes and some Rank-Metric code constructions , 2015, Adv. Math. Commun..

[44]  Ernst M. Gabidulin,et al.  The new construction of rank codes , 2005, Proceedings. International Symposium on Information Theory, 2005. ISIT 2005..

[45]  John Sheekey,et al.  MRD Codes: Constructions and Connections , 2019, 1904.05813.

[46]  John Sheekey,et al.  Rank-metric codes, linear sets, and their duality , 2018, Des. Codes Cryptogr..