Abstract
In this paper we prove a result on the structure of the elements of an additive maximum rank distance (MRD) code over the field of order two, namely that in some cases such codes must contain a semifield spread set. We use this result to classify additive MRD codes in \(M_n(\mathbb {F}_2)\) with minimum distance \(n-1\) for \(n\le 6\). Furthermore we present a computational classification of additive MRD codes in \(M_4(\mathbb {F}_3)\). The computational evidence indicates that MRD codes of minimum distance \(n-1\) are much more rare than MRD codes of minimum distance n, i.e. semifield spread sets. In all considered cases, each equivalence class has a known algebraic construction.
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Bosma W., Cannon J., Playoust C.: The Magma algebra system. I. The user language. J. Symb. Comput. 24(3–4), 235–265 (1997).
Bruen A.: Collineations and extensions of translation nets. Math. Z. 145, 243–249 (1975).
Cossidente A., Pavese F.: Subspace codes in \({\rm PG}(2n-1, q)\). Combinatorica 37, 1073–1095 (2017).
Cossidente A., Marino G., Pavese F.: Non-linear maximum rank distance codes. Des. Codes Cryptogr. 79, 597–609 (2016).
Csajbók B., Siciliano A.: Puncturing maximum rank distance codes. arXiv:1807.04108.
Csajbók B., Marino G., Polverino O., Zanella C.: A new family of MRD-codes. arXiv:1707.08487.
Csajbók B., Marino G., Polverino O., Zhou Y.: Maximum rank-distance codes with maximum left and right idealisers. arXiv:1807.08774.
Csajbók B., Marino G., Zullo F.: New maximum scattered linear sets of the projective line. arXiv:1709.00926.
de la Cruz J., Kiermaier M., Wassermann A., Willems W.: Algebraic structures of MRD codes. Adv. Math. Commun. 10, 499–510 (2016).
Delsarte P.: Bilinear forms over a finite field, with applications to coding theory. J. Comb. Theory Ser. A 25, 226–241 (1978).
Dempwolff U.: Translation planes of small order. http://www.mathematik.uni-kl.de/~dempw/dempw_Plane.html.
Dempwolff U.: Semifield planes of order 81. J. Geom. 89, 1–16 (2008).
Dempwolff U., Reifart A.: The classification of the translation planes of order 16, I. Geom. Dedicata 15, 137–153 (1983).
Dickson L.E.: On finite algebras. Nachr. Ges. Wiss. Gött. 1905, 358–393 (1905).
Dumas J.-G., Gow R., McGuire G., Sheekey J.: Subspaces of matrices with special rank properties. Linear Algebra Appl. 433, 191–202 (2010).
Etzion T., Silberstein N.: Codes and designs related to lifted MRD codes. IEEE Trans. Inform. Theory 59, 1004–1017 (2013).
Gabidulin E.M., Kshevetskiy A.: The new construction of rank codes. In: Proceedings. ISIT 2005.
Heinlein D., Honold T., Kiermaier M., Kurz S., Wassermann A.: Classifying optimal binary subspace codes of length 8, constant dimension 4 and minimum distance 6. arXiv:1711.06624.
Honold T., Kiermaier M., Kurz S.: Classification of large partial plane spreads in \({\rm PG} (6,2)\) and related combinatorial objects. arXiv:1606.07655.
Kantor W.M.: Finite Semifields, Finite Geometries, Groups, and Computation, pp. 103–114. Walter de Gruyter GmbH & Co. KG, Berlin (2006).
Knuth D.E.: Finite semifields and projective planes. J. Algebra 2, 182–217 (1965).
Lavrauw M.: Finite semifields and nonsingular tensors. Des. Codes Cryptrogr. 68, 205–227 (2013).
Lavrauw M., Polverino O.: Finite semifields. In: De Beule J., Storme L. (eds.) Current Research Topics in Galois Geometries. Nova Academic Publishers, New York (2011).
Lavrauw M., Sheekey J.: Classification of subspaces in \({\mathbb{F}}^2\otimes {\mathbb{F}}^3\) and orbits in \({\mathbb{F}}^2\otimes {\mathbb{F}}^3\otimes {\mathbb{F}}^r\). J. Geom. 108, 5–23 (2017).
Lunardon G., Trombetti R., Zhou Y.: Generalized twisted Gabidulin codes. arXiv:1507.07855.
Marino G., Polverino O.: On the nuclei of a finite semifield. Theory and applications of finite fields. Contemp. Math. 579, 123–151 (2012).
Menichetti G.: On a Kaplansky conjecture concerning three-dimensional division algebras over a finite field. J. Algebra 47, 400–410 (1977).
Ozbudak F., Otal K.: Additive rank-metric codes. IEEE Trans. Inform. Theory 63, 164–168 (2017).
Rúa I.F., Combarro E.F., Ranilla J.: Classification of semifields of order 64. J. Algebra 322, 4011–4029 (2009).
Rúa I.F., Combarro E.F., Ranilla J.: New advances in the computational exploration of semifields. Int. J. Comput. Math. 88, 1990–2000 (2011).
Rúa I.F., Combarro E.F., Ranilla J.: Determination of division algebras with 243 elements. Finite Fields Appl. 18, 1148–1155 (2012).
Rúa I.F., Combarro E.F., Ranilla J.: Finite semifields with \(7^4\) elements. Int. J. Comput. Math. 89, 1865–1878 (2012).
Schmidt K.-U., Zhou Y.: On the number of inequivalent MRD codes. arXiv:1702.04582.
Segre B.: Teoria di Galois, fibrazioni proiettive e geometrie non desarguesiane. Ann. Mat. Pura Appl. 64, 1–76 (1964).
Sheekey J.: New semifields and new MRD codes from skew polynomial rings. arXiv:1806.00251.
Sheekey J.: A new family of linear maximum rank distance codes. Adv. Math. Commun. 10, 475–488 (2016).
Silva D., Kschischang F.R., Koetter R.: A rank-metric approach to error control in random network coding. IEEE Trans. Inform. Theory 54, 3951–3967 (2008).
Trombetti R., Zhou Y.: A new family of MRD codes in \({\mathbb{F}}_{q}^{2n\times 2n}\) with right and middle nuclei \({\mathbb{F}}_{q^n}\). arXiv:1709.03908.
Walker R.J.: Determination of division algebras with \(32\) elements. In: Proceedings of Symposia in Applied Mathematics, American Mathematical Society, pp. 83–85 (1962).
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Sheekey, J. Binary additive MRD codes with minimum distance \(n-1\) must contain a semifield spread set. Des. Codes Cryptogr. 87, 2571–2583 (2019). https://doi.org/10.1007/s10623-019-00637-6
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DOI: https://doi.org/10.1007/s10623-019-00637-6