×

Flag codes from planar spreads in network coding. (English) Zbl 1464.94081

Summary: In this paper we study a class of multishot network codes given by families of nested subspaces (flags) of a vector space \(\mathbb{F}_q^n\), being \(q\) a prime power and \(\mathbb{F}_q\) the finite field of \(q\) elements. In particular, we focus on flag codes having maximum distance (optimum distance flag codes). We explore the existence of these codes from spreads, based on the good properties of the latter ones. For \(n=2k\), we show that optimum distance full flag codes with the largest size are exactly those that can be constructed from a planar spread. We give a precise construction of them as well as a decoding algorithm.

MSC:

94B35 Decoding
94B60 Other types of codes
51E99 Finite geometry and special incidence structures
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Ahlswede, R.; Cai, N.; Li, R.; Yeung, R. W., Network information flow, IEEE Trans. Inf. Theory, 46, 1204-1216 (2000) · Zbl 0991.90015
[2] Etzion, T.; Raviv, N., Equidistant codes in the Grassmannian, Discrete Appl. Math., 186, 87-97 (2015) · Zbl 1384.94136
[3] Gorla, E.; Ravagnani, A., Partial spreads in random network coding, Finite Fields Appl., 26, 104-115 (2014) · Zbl 1288.94107
[4] Gorla, E.; Ravagnani, A., Equidistant subspace codes, Linear Algebra Appl., 490, 48-65 (2016) · Zbl 1401.94253
[5] Koetter, R.; Kschischang, F., Coding for errors and erasures in random network coding, IEEE Trans. Inf. Theory, 54, 3579-3591 (2008) · Zbl 1318.94111
[6] Lavrauw, M., Scattered Spaces with respect to Spreads and Eggs in Finite Projective Spaces (2001), Eindhoven University of Technology: Eindhoven University of Technology Eindhoven, PhD Dissertation · Zbl 0990.51003
[7] Lavrauw, M.; Van de Voorde, G., Field Reduction and Linear Sets in Finite Geometry, Contemporary Mathematics, vol. 632, 271-293 (2015), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 1351.51008
[8] Liebhold, D.; Nebe, G.; Vazquez-Castro, A., Network coding with flags, Des. Codes Cryptogr., 86, 2, 269-284 (2018) · Zbl 1412.94254
[9] Manganiello, F.; Gorla, E.; Rosenthal, J., Spread codes and spread decoding in network coding, (IEEE International Symposium on Information Theory. IEEE International Symposium on Information Theory, Proceedings (ISIT), Toronto, Canada (2008)), 851-855
[10] Manganiello, F.; Trautmann, A.-L., Spread decoding in extension fields, Finite Fields Appl., 25, 94-105 (2014) · Zbl 1305.94026
[11] Nóbrega, R. W.; Uchôa-Filho, B. F., Multishot codes for network coding: bounds and a multilevel construction, (2009 IEEE International Symposium on Information Theory. 2009 IEEE International Symposium on Information Theory, Proceedings (ISIT), Seoul, South Korea (2009)), 428-432
[12] Nóbrega, R. W.; Uchôa-Filho, B. F., Multishot codes for network coding using rank-metric codes, (2010 Third IEEE International Workshop on Wireless Network Coding. 2010 Third IEEE International Workshop on Wireless Network Coding, Boston, USA (2010)), 1-6
[13] Segre, B., Teoria di Galois, Fibrazioni Proiettive e Geometrie non Desarguesiane, Ann. Mat. Pura Appl., 64, 1-76 (1964) · Zbl 0128.15002
[14] Trautmann, A.-L.; Rosenthal, J., Constructions of constant dimension codes, (Greferath, M.; etal., Network Coding and Subspace Designs (2018), E-Springer International Publishing AG), 25-42
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.