Topic
Cyclic subspace
About: Cyclic subspace is a research topic. Over the lifetime, 65 publications have been published within this topic receiving 1095 citations.
Papers published on a yearly basis
Papers
TL;DR: Several new bounds on the size of codes in Pq(n), which may be thought of as counterparts of the classical bounds in coding theory due to Johnson, Delsarte, and Gilbert-Varshamov are presented.
Abstract: The projective space of order n over the finite field \BBFq, denoted here as Pq(n), is the set of all subspaces of the vector space \BBFqn . The projective space can be endowed with the distance function d(U, V) = dimU + dimV -2 dim(U ∩ V) which turns Pq(n) into a metric space. With this, an (n,M,d) code \BBC in projective space is a subset of Pq(n) of size M such that the distance between any two codewords (subspaces) is at least d . Koetter and Kschischang recently showed that codes in projective space are precisely what is needed for error-correction in networks: an (n,M,d) code can correct t packet errors and ρ packet erasures introduced (adversarially) anywhere in the network as long as 2t + 2ρ <; d. This motivates our interest in such codes. In this paper, we investigate certain basic aspects of “coding theory in projective space.” First, we present several new bounds on the size of codes in Pq(n), which may be thought of as counterparts of the classical bounds in coding theory due to Johnson, Delsarte, and Gilbert-Varshamov. Some of these are stronger than all the previously known bounds, at least for certain code parameters. We also present several specific constructions of codes and code families in Pq(n). Finally, we prove that nontrivial perfect codes in Pq(n) do not exist.
334 citations
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TL;DR: In this paper, the authors show how the structure of cyclic orbit codes can be used to compute the minimum distance and cardinality of a given code and propose different decoding procedures for a particular subclass.
Abstract: A constant dimension code consists of a set of k-dimensional subspaces of \BBF qn. 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 \BBF qn. If the acting group is cyclic, the corresponding orbit codes are called cyclic orbit codes. In this paper, we show how orbit codes can be seen as an analog of linear codes in the block coding case. We investigate how the structure of cyclic orbit codes can be utilized to compute the minimum distance and cardinality of a given code and propose different decoding procedures for a particular subclass of cyclic orbit codes.
115 citations
TL;DR: In this article, the cardinality of the cyclic subspace codes is determined using the largest subfield over which the given subspace is a vector space, and estimates for its distance can be found.
Abstract: Cyclic orbit codes are constant dimension subspace codes that arise as the orbit of a
cyclic subgroup of the general linear group acting on subspaces in the given ambient space.
With the aid of the largest subfield over which the given subspace is a vector space, the cardinality
of the orbit code can be determined, and estimates for its distance can be found.
This subfield is closely related to the stabilizer of the generating subspace.
Finally, with a linkage construction larger, and longer, constant dimension codes can be derived from cyclic
orbit codes without compromising the distance.
71 citations
TL;DR: This paper considers cyclic subspace codes, which are cyclic and provides constructions of optimal codes for which their codewords do not have full orbits, and introduces a new way to represent sub space codes by a class of polynomials called subspace polynOMials.
Abstract: Subspace codes have received an increasing interest recently due to their application in error correction for random network coding. In particular, cyclic subspace codes are possible candidates for large codes with efficient encoding and decoding algorithms. In this paper, we consider such cyclic codes and provide constructions of optimal codes for which their codewords do not have full orbits. We further introduce a new way to represent subspace codes by a class of polynomials called subspace polynomials. We present some constructions of such codes, which are cyclic and analyze their parameters.
70 citations
TL;DR: In this article, a set of constructions of non-trivial cyclic subspace codes in which the relation between $k$ and $n$ is polynomial and in particular linear is given.
Abstract: A subspace of a finite extension field is called a Sidon space if the product of any two of its elements is unique up to a scalar multiplier from the base field. Sidon spaces were recently introduced by Bachoc et al. as a means to characterize multiplicative properties of subspaces, and yet no explicit constructions were given. In this paper, several constructions of Sidon spaces are provided. In particular, in some of the constructions the relation between $k$ , the dimension of the Sidon space, and $n$ , the dimension of the ambient extension field, is optimal. These constructions are shown to provide cyclic subspace codes, which are useful tools in network coding schemes. To the best of our knowledge, this constitutes the first set of constructions of non-trivial cyclic subspace codes in which the relation between $k$ and $n$ is polynomial, and in particular, linear. As a result, a conjecture by Trautmann et al. regarding the existence of non-trivial cyclic subspace codes is resolved for most parameters, and multi-orbit cyclic subspace codes are attained, whose cardinality is within a constant factor (close to 1/2) from the sphere-packing bound for subspace codes.
62 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2022 | 1 |
| 2021 | 6 |
| 2020 | 2 |
| 2019 | 2 |
| 2018 | 6 |
| 2017 | 5 |