Message encoding and retrieval for spread and cyclic orbit codes

Abstract

Spread codes and cyclic orbit codes are special families of constant dimension subspace codes. These codes have been wellstudied for their error correction capability, transmission rate and decoding methods, but the question of how to encode and retrieve messages has not been investigated. In this work we show how a message set of consecutive integers can be encoded and retrieved for these two code families.

Flag Codes: Distance Vectors and Cardinality Bounds

“…The study and construction of constant dimension codes attaining this upper bound for the distance has been addressed in several papers (see [8,18], for instance). Another important problem is the one of determining (or giving bounds for) the value A q (n, d, k), which denotes the maximum possible size for constant dimension codes in G q (k, n) having prescribed minimum distance d. The reader can find constructions of constant dimension codes as well as lower and upper bounds for A q (n, d, k) in [6,10,12,13,16,20,21,22,23,24]. As a generalization of constant dimension codes, in [17], the authors introduced the use of flag codes in network Coding.…”

“…, d S (C r )). Optimum distance flag codes in F q (t, n) are a particular class of consistent flag codes whose associated distance vector is D (t,n) defined in (12).…”

“…According to this, there are two central problems when working with constant dimension codes. On the one hand, the study and construction of codes having the maximum possible distance for their dimension (see, for instance, [7,8,12,18]). On the other hand, determining (or giving bounds for) the value A q (n, d, k), that is, the maximum possible size of a constant dimension code in G q (k, n) with minimum distance equal to d, is an interesting question that has led to many research works (see [10,13,16,24], for instance).…”

Symbol Erasure Correction in Random Networks With Spread Codes

“…Proof. In [4,Lemma 17] it is shown that the map φ can be carried out in O(k 2 ) operations over the field F q . Since this needs to be done for any of the m blocks of the codeword matrix representation, the statement follows.…”

Orbit codes from forms on vector spaces over a finite field

“…By [44,Proposition 3.11], if a code has G as a generating subgroup, then G is a subgroup of the automorphism group of the code. These codes were introduced in [46], and since then they have been further investigated by many authors [8,30,45,40]. It is well known that GL(V ) contains exactly one conjugacy class of cyclic subgroups, acting regularly on V \ {0} and isomorphic to F q n \ {0}, i.e., the Singer groups.…”