Further constructions of cyclic subspace codes
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Further constructions of cyclic subspace codes He Zhang1 · Xiwang Cao1,2 Received: 19 July 2018 / Accepted: 13 October 2020 / © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Subspace codes, especially cyclic subspace codes, have attracted a wide attention in the past few decades due to their applications in error correction for random network coding. In 2016, Ben-Sasson et al. gave a systematic approach to constructing cyclic subspace codes by employing subspace polynomials. Inspired by Ben-Sasson’s idea, Chen et al. also provided some constructions of cyclic subspace codes in 2017. In this paper, two constructions of cyclic subspace codes are given to further improve the results of Chen and Roth et al. respectively. Consequently, we obtain more cyclic subspace codes with larger size of codewords without reducing the minimum distance. Keywords Constant dimension code · Subspace polynomial · Random network coding · Linearized polynomial · Subspace code · Sidon space Mathematics Subject Classification (2010) 11T71 · 11T06
1 Introduction Let q be a prime power and N a positive integer. Let Fq be a finite field with q elements and Fq N an extension of Fq . Let FN q denote the vector space over Fq of dimension N . It is known that Fq N can be regarded as a vector space over Fq of dimension N . Here and after, we will not distinguish between Fq N and FN q as vector spaces. Denote by Pq (N ) the set of all subspaces of Fq N . For each positive integer k ≤ N , let Gq (N, k) stand for the set of all subspaces of Fq N with dimension k, i.e. the k-Grassmannian. We denote by
This work was supported by the National Natural Science Foundation of China (Grant Nos. 11771007). Xiwang Cao
[email protected] He Zhang [email protected] 1
Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, 211100, China
2
State Key Laboratory of Information Security, Institute of Information Engineering, Chinese Academy of Sciences, Beijing, 100093, China
Cryptography and Communications
k−1 q N −i −1 N = the q-ary Gaussian coefficient, which is the number of Gq (N, k). It is k q i=0 q k−1 −1 obvious that N Pq (N ) = Gq (N, k).
k=0
For any U , V ∈ Pq (N ), the subspace distance between U and V is defined by V ). d(U, V ) = dim(U ) + dim(V ) − 2 dim(U The Pq (N ) space equipped with the metric d becomes a metric space. Any nonempty subset C of Gq (N, k) is termed a constant dimension code (or k-dimensional subspace code). For any subspace code C ⊂ Gq (N, k), its minimum distance, which is denoted by d(C ), is defined as d(C ) = min {d(U, V ) : U, V ∈ C , U = V } .
Let F∗q N = Fq N \{0}, V ∈ Gq (N, k) and α ∈ F∗q N . A cyclic shift of V is represented as αV = {αv|v ∈ V }. It is clear from the definition that αV is a vector space over Fq with the same dimension as V . Two cyclic shifts of V are called distinct if they form two different vector spaces. A k-dimensional subspace code C ⊆ Gq (N, k) is considered as cyclic if αV ∈ C for all α ∈ F∗q N and V ∈ C . T
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