Several kinds of large cyclic subspace codes via Sidon spaces

https://doi.org/10.1016/j.disc.2019.111788Get rights and content

Abstract

Subspace codes have attracted much attention in recent years due to their applications to error correction in random network coding. In this paper, we construct several kinds of large cyclic subspace codes via Sidon spaces and large subspace codes via unions of some Sidon spaces. Therefore, some known results are extended.

Introduction

Let Fq be a finite field with q elements and Fqn be an extension field of degree n over Fq. The projective space of order n over Fq (see [7]), denoted by Pq(n), is the set of all Fq-subspaces of Fqn. For any U,VPq(n), the subspace distance of U, V is defined as: d(U,V)=dim(U+V)dim(UV)=dim(U)+dim(V)2dim(UV),where dim() denotes the dimension of a vector space over Fq. A nonempty subset C of Pq(n) with the subspace distance is called a subspace code. The minimum subspace distance of a subspace code C is defined as: d(C)=min{d(U,V)|U,VC,UV}.We denote by Gq(n,k) the set of all k-dimensional subspaces of Fqn. A subspace code C is termed constant dimension code if all the elements of C have the same dimension k, i.e., C is a subset of Gq(n,k).

In order to construct subspace codes with a larger code size and a large minimum subspace distance for a fixed q,n, and k, we introduce cyclic subspace codes. For αFqn and UGq(n,k), αU={αu|uU} is called the cyclic shift of U. It is easy to see that αU and U have the same dimension. A subspace code C is called cyclic if αUC for all αFqn and UGq(n,k).

Subspace codes were defined in [11] as a tool studying random network coding, which is first introduced in [1]. Cyclic subspace codes and especially cyclic constant dimension codes are very useful with efficient encoding and decoding algorithms [8]. Soon afterwards, subspace codes received a great deal of attention and research (see [6], [12], [15]). A lot of efforts have been devoted to the study of cyclic subspace codes.

  • (1)

    Ben-Sasson et al. [3] used the set of roots of the subspace polynomial xqk+xq+xFq[x] to denote the subspace U and shown that {αU|αFqn} is a cyclic subspace code in Gq(n,k) for any given k and infinitely many values of n. In addition, they provided a method to increase the number of distinct cyclic subspace code without decreasing the minimum subspace distance.

  • (2)

    Otal and Özbudak [14] improved the construction in [3] by the roots of xqk+θixq+γixFqm[x] and obtained several kinds of cyclic subspace codes.

  • (3)

    Chen and Liu [4] generalized the previous works [3] and [14] by a different approach. They studied the roots of the trinomials xqk+alxql+a0xFqm[x], where 1l<k,gcd(l,k)=1 and a0,alFqn.

  • (4)

    Zhao and Tang [18] provided a new way to research the roots of a quadrinomial and presented a kind of new cyclic subspace codes.

  • (5)

    Roth et al. [16] putted forward a seminal work. They presented a novel construction of cyclic subspace codes through Sidon spaces. They proved that constructing a cyclic subspace code in Gq(n,k) with size qn1q1 and minimum subspace distance 2k2 is equivalent to constructing a Sidon space in Gq(n,k) (a fact which is also shown in [2]) and exhibited several kinds of methods to construct Sidon spaces.

In this paper, we explore the ideas of constructing Sidon spaces proposed in [16]. As a consequence, several new kinds of Sidon spaces are obtained. This paper is organized as follows. In Section 2, we give some basic results for obtaining our main results. In Section 3, we extend the previous works and construct several kinds of Sidon spaces. Furthermore, unions of cyclic subspace codes from Sidon spaces are also discussed. In Section 4, we conclude the paper.

Section snippets

Preliminaries

In this section, we will review some basic results. Throughout this paper, Fq denotes the finite field with q elements and Fqn denotes the extension field of degree n over Fq, where n1 is a positive integer. Recall that Gq(n,k) denotes the set of all k-dimensional Fq-subspaces of Fqn.

Consider the action of the multiplicative group Fqn to the set Gq(n,k). For each UGq(n,k), define a cyclic subspace code as follows: C={αU|αFqn}.In order to compute the size of C, set H={βFqn|βU=U}the

Main results

In this section, we construct several kinds of cyclic subspace codes via Sidon spaces and large subspace codes via unions of some Sidon spaces.

Theorem 3.1

For two positive integers k and n withkn, letf(x)Fqk[x] be an irreducible polynomial with degree nk>5 and γFqn be a root of f(x). Then U={a+uγ+uqγ2:uFqk,aFq}Gq(n,k+1)is a Sidon space.

Proof

Now we check whether U is a Sidon space by Definition 2.2.

Let a+uγ+uqγ2,b+vγ+vqγ2,a+uγ+uqγ2,b+vγ+vqγ2be four non-zero elements in U such that (a+uγ+uqγ2)(b+vγ+vq

Concluding remarks

In this paper, we presented several new kinds of cyclic subspace codes by employing Sidon spaces and unions of some Sidon spaces. Moreover, some cyclic subspace codes presented in this paper have size exceeding qn1q1 and minimum subspace distance still remain exactly 2k2.

Declaration of Competing Interest

The authors declare that there are no known conflicts of interest associated with this publication and there has been no significant financial support for this work that could have influenced its outcome.

References (18)

  • ZhaoW. et al.

    A characterization of cyclic subspace codes via subspace polynomials

    Finite Fields Appl.

    (2019)
  • AhlswedeR. et al.

    Network information flow

    IEEE Trans. Inform. Theory

    (2000)
  • BachocC. et al.

    An analogue of Vosper’s theorem for extension fields

    Math. Proc. Cambridge Philos. Soc.

    (2017)
  • Ben-SassonE. et al.

    Subspace polynomials and cyclic subspace codes

    IEEE Trans. Inform. Theory

    (2016)
  • ChenB. et al.

    Constructions of cyclic constant dimension codes

    Des. Codes Cryptogr.

    (2017)
  • DixonD. et al.

    Permutation Groups

    (1996)
  • EtzionT. et al.

    Error-correcting codes in projective spaces via rank-metric codes and ferrers diagrams

    IEEE Trans. Inform. Theory

    (2009)
  • EtzionT. et al.

    Error-correcting codes in projective space

    IEEE Trans. Inform. Theory

    (2011)
  • GabidulinE. et al.

    Decoding of random network codes

    Probl. Inf. Transm.

    (2010)
There are more references available in the full text version of this article.

Cited by (21)

  • Multi-orbit cyclic subspace codes and linear sets

    2023, Finite Fields and their Applications
  • Construction of constant dimension codes via improved inserting construction

    2023, Applicable Algebra in Engineering, Communications and Computing
View all citing articles on Scopus

This work was supported in part by National Natural Science Foundation of China (Nos. 61772015, 11661014), the Foundation of Science and Technology on Information Assurance Laboratory, China under Grant KJ-17-010.

View full text