Spread Decoding in Extension Fields

A spread code is a set of vector spaces of a fixed dimension over a finite field Fq with certain properties used for random network coding. It can be constructed in different ways which lead to different decoding algorithms. In this work we present a new representation of spread codes with a minimum distance decoding algorithm which is efficient when the codewords, the received space and the error space have small dimension.Comment: Submitted for publication to Finite Fields and their Applications (Elsevier

Message Encoding for Spread and Orbit Codes

Spread codes and orbit codes are special families of constant dimension subspace codes. These codes have been well-studied for their error correction capability and transmission rate, but the question of how to encode messages has not been investigated. In this work we show how the message space can be chosen for a given code and how message en- and decoding can be done.Comment: Submitted to IEEE International Symposium on Information Theory 201

Isometry and Automorphisms of Constant Dimension Codes

We define linear and semilinear isometry for general subspace codes, used for random network coding. Furthermore, some results on isometry classes and automorphism groups of known constant dimension code constructions are derived

Cyclic Orbit Codes

In network coding a constant dimension code consists of a set of k-dimensional subspaces of F_q^n. Orbit codes are constant dimension codes which are defined as orbits of a subgroup of the general linear group, acting on the set of all subspaces of F_q^n. If the acting group is cyclic, the corresponding orbit codes are called cyclic orbit codes. In this paper we give a classification of cyclic orbit codes and propose a decoding procedure for a particular subclass of cyclic orbit codes.Comment: submitted to IEEE Transactions on Information Theor

A complete characterization of irreducible cyclic orbit codes and their Plücker embedding

Constant dimension codes are subsets of the finite Grassmann variety. The study of these codes is a central topic in random linear network coding theory. Orbit codes represent a subclass of constant dimension codes. They are defined as orbits of a subgroup of the general linear group on the Grassmannian. This paper gives a complete characterization of orbit codes that are generated by an irreducible cyclic group, i.e. a group having one generator that has no non-trivial invariant subspace. We show how some of the basic properties of these codes, the cardinality and the minimum distance, can be derived using the isomorphism of the vector space and the extension field. Furthermore, we investigate the Plücker embedding of these codes and show how the orbit structure is preserved in the embeddin

Flag codes from planar spreads in network coding

In this paper we study a class of multishot network codes given by families of nested subspaces (flags) of a vector space Fnq, being qa prime power and Fqthe finite field of qelements. 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.The first and third authors are partially supported by Projecte AICO/2017/128 of Generalitat Valenciana. The second author is supported by Generalitat Valenciana and Fondo Social Europeo. Grants: ACIF/2018/196 and BEFPI/2019/070. The third author is also supported by the grant BEST/2019/192 of Generalitat Valenciana

Optimum distance flag codes from spreads via perfect matchings in graphs

In this paper, we study flag codes on the vector space Fnq, being q a prime power and Fq the finite field of q elements. More precisely, we focus on flag codes that attain the maximum possible distance (optimum distance flag codes) and can be obtained from a spread of Fnq. We characterize the set of admissible type vectors for this family of flag codes and also provide a construction of them based on well-known results about perfect matchings in graphs. This construction attains both the maximum distance for its type vector and the largest possible cardinality for that distance.Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature. The authors receive financial support from Ministerio de Ciencia e Innovación PID2019-108668GB-I00 (Spain). The first and third authors are partially supported by Projecte AICO/2017/128 of Generalitat Valenciana (Spain). The second author is supported by Generalitat Valenciana and Fondo Social Europeo, grant number: ACIF/2018/196 (Spain)

On Generalized Galois Cyclic Orbit Flag Codes

Flag codes that are orbits of a cyclic subgroup of the general linear group acting on flags of a vector space over a finite field, are called cyclic orbit flag codes. In this paper, we present a new contribution to the study of such codes, by focusing this time on the generating flag. More precisely, we examine those ones whose generating flag has at least one subfield among its subspaces. In this situation, two important families arise: the already known Galois flag codes, in case we have just fields, or the generalized Galois flag codes in other case. We investigate the parameters and properties of the latter ones and explore the relationship with their underlying Galois flag code.This research was funded by Ministerio de Ciencia e Innovación (grant number PID2019-108668GB-I00) and Generalitat Valenciana y Fondo Social Europeo (Grant number ACIF/2018/196)