Top

Designs, Codes and Cryptography

Published in:

08-07-2017

A generalization of the normal rational curve in \(\mathop {\mathrm{PG}}(d,q^n)\) and its associated non-linear MRD codes

Authors: Giorgio Donati, Nicola Durante

Published in: Designs, Codes and Cryptography | Issue 6/2018

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

Abstract

Let A and B be two points of \(\mathop {\mathrm{PG}}(d,q^n)\) and let \(\Phi \) be a collineation between the stars of lines with vertices A and B, that does not map the line AB into itself. In this paper we prove that if \(d=2\) or \(d\ge 3\) and the lines \(\Phi ^{-1}(AB), AB, \Phi (AB) \) are not in a common plane, then the set \(\mathcal{C}\) of points of intersection of corresponding lines under \(\Phi \) is the union of \(q-1\) scattered \({\mathbb {F}}_{q}\)-linear sets of rank n together with \(\{A,B\}\). As an application we will construct, starting from the set \(\mathcal{C}\), infinite families of non-linear \((d+1, n, q;d-1)\)-MRD codes, \(d\le n-1\), generalizing those recently constructed in Cossidente et al. (Des Codes Cryptogr 79:597–609, 2016) and Durante and Siciliano (Electron J Comb, 2017).

next article A recursive construction for simple t-designs using resolutions

Blokhuis A., Lavrauw M.: Scattered spaces with respect to a spread in PG(\(n, q\)). Geom. Dedicata 81, 231–243 (2000).MathSciNetCrossRefMATH

Clifford W.K.: On the classification of loci. Philos. Trans. R. Soc. II, 663–681 (1878).CrossRef

Cooperstein B.N.: External flats to varieties in PG(\(M_{n, n}\)GF(\(q\))). Linear Algebra Appl. 267, 175–186 (1997).MathSciNetMATH

Cossidente A., Marino G., Pavese F.: Non-linear maximum rank distance codes. Des. Codes Cryptogr. 79, 597–609 (2016).MathSciNetCrossRefMATH

Davis K.S., Webb W.A.: Lucas’ theorem for prime powers. Eur. J. Comb. 11, 229–233 (1990).MathSciNetCrossRefMATH

Delsarte P.: Bilinear forms over a finite field, with applications to coding theory. J. Comb. Theory Ser. A 25, 226–241 (1978).MathSciNetCrossRefMATH

Dembowski P.: Finite Geometries. Springer, New York (1968).CrossRefMATH

Donati G.: On the system of fixed points of a collineation in non commutative projective geometry. Discret. Math. 255, 65–70 (2002).CrossRefMATH

Donati G., Durante N.: Scattered linear sets generated by collineations between pencils of lines. J. Algebraic Comb. 40, 1121–1134 (2014).MathSciNetCrossRefMATH

10.

Durante N., Siciliano A.: Non-linear maximum rank distance codes in the cyclic model for the field reduction of finite geometries. Electron. J. Comb. (2017). arXiv:1704.02110.

11.

Gabidulin E.M.: Theory of codes with maximum rank distance. Probl. Pereda. Inf. 21, 3–16 (1985).MathSciNetMATH

12.

Hirschfeld J.W.P., Thas J.A.: General Galois Geometries, 2nd edn. Springer, New York (1991).MATH

13.

Kshevetskiy A., Gabidulin E.M.: The new construction of rank codes. In; Proceedings IEEE International Symposium on Information Theory, pp. 2105–2108 (2005).

14.

Lunardon G.: MRD-codes and linear sets. J. Comb. Theory Ser. A 149, 1–20 (2017).MathSciNetCrossRefMATH

15.

Lunardon G., Polverino O.: Translation ovoids of orthogonal polar spaces. Forum. Math. 16, 663–669 (2004).MathSciNetCrossRefMATH

16.

Lunardon G., Marino G., Polverino O., Trombetti R.: Maximum scattered linear sets of pseudoregulus type and the Segre Variety \({\cal{S}}_{n, n}\). J. Algebra Comb. 39, 807–831 (2014).MathSciNetCrossRefMATH

17.

Lunardon G., Trombetti R., Zhou Y.: Generalized twisted Gabidulin codes (submitted).

18.

Polverino O.: Linear sets in finite projective spaces. Discret. Math. 310, 3096–3107 (2010).MathSciNetCrossRefMATH

19.

Seydewitz F.: Lineäre Konstruktion einer Kurve doppelter Krümmung. Arch. Math. Phys. 10, 203–214 (1847).

20.

Sheekey J.: A new family of linear maximum rank distance codes. Adv. Math. Commun. 10, 475–488 (2016).MathSciNetCrossRefMATH

21.

Veronese G.: Behandlung der projectivischen Verhältnisse der Räume von verschiedenen dimensionen durch das Princip des Projicirens und Schneidens. Math. Ann. 19, 161–234 (1882).CrossRefMATH

Title: A generalization of the normal rational curve in and its associated non-linear MRD codes
Authors: Giorgio Donati
Nicola Durante
Publication date: 08-07-2017
Publisher: Springer US
Published in: Designs, Codes and Cryptography / Issue 6/2018
Print ISSN: 0925-1022
Electronic ISSN: 1573-7586
DOI: https://doi.org/10.1007/s10623-017-0388-7

Springer Professional

A generalization of the normal rational curve in \(\mathop {\mathrm{PG}}(d,q^n)\) and its associated non-linear MRD codes

Abstract

Premium Partner

Springer Professional

Abstract

Please log in to get access to your license.

Other articles of this Issue 6/2018

A framework for constructing partial geometric difference sets

Miklós–Manickam–Singhi conjectures on partial geometries

On two-weight -codes

A construction of group divisible designs with block sizes 3 to 7

A recursive construction for simple t-designs using resolutions

Leakage-resilient attribute based encryption in prime-order groups via predicate encodings

Premium Partner