Geometrical Aspects of Subspace Codes

A subspace code of size <inline-formula><tex-math id="M1">\begin{document}$ \bf{333} $\end{document}</tex-math></inline-formula> in the setting of a binary <inline-formula><tex-math id="M2">\begin{document}$ \bf{q} $\end{document}</tex-math></inline-formula>-analog of the Fano plane

“…the group G 16,1 performs pretty bad and the LP relaxation gives an upper bound of 1292. Over the ternary field the first open case is 754 ≤ A 3 (6, 4; 3) ≤ 784, see [25,Theorem 2] or [13,12]. Using the systematic approach we were able to reproduce the best known size 754, but unfortunately no improvement above that has been found.…”

Orbit codes from forms on vector spaces over a finite field

“…By putting i = 0 and i = k + 1 in the first and in the second equation of (14), respectively, we get g k = g k+1 = 0. Hence, from (13) and 14, we have g m = 0 for m = 1, . .…”

“…In view of their application in random network coding, subspace codes have been intensely studied in recent years (see for instance [31,39,13,29] and references therein). One of the main problems of subspace coding asks for the maximum possible cardinality of a subspace code of length n over F q with minimum distance at least d and the classification of the corresponding optimal codes.…”

Combining subspace codes

“…We will generalize this approach in Lemma 4.1. Finally, for constructions obtained by using geometrical techniques we refer the interested reader to [5,6,7,8,9].…”