Several kinds of large cyclic subspace codes via Sidon spaces☆
Introduction
Let be a finite field with elements and be an extension field of degree over . The projective space of order over (see [7]), denoted by , is the set of all -subspaces of . For any , the subspace distance of , is defined as: where denotes the dimension of a vector space over . A nonempty subset of with the subspace distance is called a subspace code. The minimum subspace distance of a subspace code is defined as: We denote by the set of all -dimensional subspaces of . A subspace code is termed constant dimension code if all the elements of have the same dimension , i.e., is a subset of .
In order to construct subspace codes with a larger code size and a large minimum subspace distance for a fixed , and , we introduce cyclic subspace codes. For and , is called the cyclic shift of . It is easy to see that and have the same dimension. A subspace code is called cyclic if for all and .
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 to denote the subspace and shown that is a cyclic subspace code in for any given and infinitely many values of . In addition, they provided a method to increase the number of distinct cyclic subspace code without decreasing the minimum subspace distance.
- (2)
Otal and zbudak [14] improved the construction in [3] by the roots of and obtained several kinds of cyclic subspace codes.
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Chen and Liu [4] generalized the previous works [3] and [14] by a different approach. They studied the roots of the trinomials , where and .
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Zhao and Tang [18] provided a new way to research the roots of a quadrinomial and presented a kind of new cyclic subspace codes.
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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 with size and minimum subspace distance is equivalent to constructing a Sidon space in (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, denotes the finite field with elements and denotes the extension field of degree over , where is a positive integer. Recall that denotes the set of all -dimensional -subspaces of .
Consider the action of the multiplicative group to the set . For each , define a cyclic subspace code as follows: In order to compute the size of , set 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 and with, let be an irreducible polynomial with degree and be a root of . Then is a Sidon space.
Proof Now we check whether is a Sidon space by Definition 2.2. Let be four non-zero elements in such that
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 and minimum subspace distance still remain exactly .
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.
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