Cyclic orbit flag codes

Abstract

In network coding, a flag code is a set of sequences of nested subspaces of F n q , being F q the finite field with q elements. Flag codes defined as orbits of a cyclic subgroup of the general linear group acting on flags of F n q are called cyclic orbit flag codes. Inspired by the ideas in [10], we determine the cardinality of a cyclic orbit flag code and provide bounds for its distance with the help of the largest subfield over which all the subspaces of a flag are vector spaces (the best friend of the flag). Special attention is paid to two specific families of cyclic orbit flag codes attaining the extreme possible values of the distance: Galois cyclic orbit flag codes and optimum distance cyclic orbit flag codes. We study in detail both classes of codes and analyze the parameters of the respective subcodes that still have a cyclic orbital structure.

Construction and bounds for subspace codes

“…Referring to [146] the authors of [142] state that the Singleton bound is always stronger than the sphere packing bound for non-trivial codes. However, for = 2, = 8, = 6, and = 4, the sphere-packing bound gives an upper bound of 200787/451 ≈ 445.20399 while the Singleton bound gives an upper bound of 6 4 2 = 651. For = 2, = 8, = 4, and = 4 it is just the other way round, i.e., the Singleton bound gives 7 3 2 = 11811 and the sphere-packing bound gives 8 4 2 = 200787.…”

“…In e.g. [154] (6,4; 3) = 77 and 2 (8, 6; 4) = 257, which are obtained via extensive ILP computations, see [132] and [119], respectively.…”

“…Let W be a (12, 6; 4) -CDC constructed via the improved linkage construction in Theorem 5.12 with = 6. Determine all v ∈ G 1 (12,6) such that for every ∈ G (12, 6) with pivot vector v we have S (W, ) ≥ 4.…”

“…To this end, let C 1 = G (6, 4), C 2 = G (6, 2), and M be a (4×4, 2) -MRD code. For = 6 2 / (6, 4; 2) = 5 1 let C 1 2 , . .…”

Submodule codes as spherical codes in buildings

“…For example, in the applications of flags to network coding, spherical buildings are used. Such strategy, first introduced in [23], has found further developments in [4,5,6,22] and variations in [16]. Moreover, Bruhat-Tits buildings also make their appearance in the study of holographic codes [25] as well as in the study of valued rank-metric codes [12].…”