Constant Dimension Codes Research Articles

Parallel construction for constant dimension codes from mixed dimension construction

Parallel construction for constant dimension codes from mixed dimension construction

Lifted Codes with Construction of Echelon-Ferrers for Constant Dimension Codes

Finding the highest possible cardinality, Aq(n,d;k), of the set of k-dimensional subspaces in Fqn, also known as codewords, is a fundamental problem in constant dimension codes (CDCs). CDCs find applications in a number of domains, including distributed storage, cryptography, and random linear network coding. The goal of recent research papers has been to establish Aq(n,d;k). We further improved the echelon-Ferrers construction with an algorithm, and enhanced the inserting construction by swapping specified columns of the generator matrix to obtain new lower bounds.

Large optimal cyclic subspace codes

Constant dimension codes, a specialized subclass of subspace codes, have garnered significant attention in recent years due to their applications in random network coding. Among these codes, large cyclic constant dimension codes, endowed with optimal minimum distance, offer efficient solutions for encoding and decoding algorithms. This paper is dedicated to the design of cyclic constant dimension codes that achieve the dual goals: maximizing code size while maintaining optimal minimum distance. We first explore a new form of Sidon spaces and employ this new form to construct several families of cyclic subspace codes. Our codes have optimal minimum distances and have more codewords than previous constructions in the literature.

Equivalence for flag codes

Given a finite field Fq and a positive integer n, a flag is a sequence of nested Fq-subspaces of a vector space Fqn and a flag code is a nonempty collection of flags. The projected codes of a flag code are the constant dimension codes containing all the subspaces of prescribed dimensions that form the flags in the flag code.In this paper we address the notion of equivalence for flag codes and explore in which situations such an equivalence can be reduced to the equivalence of the corresponding projected codes. In addition, this study leads to new results concerning the automorphism group of certain families of flag codes, some of them also introduced in this paper.

New constructions of Sidon spaces and large cyclic constant dimension codes

New constructions of Sidon spaces and large cyclic constant dimension codes

Constant dimension codes from multilevel construction based on matchings of complete hypergraphs

Constant dimension codes (CDCs), as special subspace codes, have received a lot of attention due to its application in random network coding. Multilevel construction and its generalizations play a key role in the constructions of CDCs. In this paper, we show a new generalization of multilevel construction based on matchings of complete hypergraphs. Some lower bounds of Aq(n,2δ,k), especially k<2δ, can be improved by applying this construction.

New constructions of constant dimension codes by improved inserting construction

Let C be a constant dimension code whose codewords are k-dimensional subspaces of Fqm. One of the main research problems of constant dimension codes is to calculate the maximum possible cardinality Aq(m,d,k) of the code whose the distance between any two different codewords is at least d. In this paper, we propose a new construction approach of constant dimension codes. The constant dimension codes based on this construction can be inserted into the parallel linkage construction. Moreover, the proposed construction gives new lower bounds of Aq(m,d,k) for m=2k. Some constant dimension codes with larger cardinality than the previously best known codes are given.

Double Multilevel Constructions for Constant Dimension Codes

Abstract: Constant dimension codes (CDCs), as special subspace codes, have received a lot of attention due to their application in random network coding. This paper introduces a family of new codes, called Ferrers diagram rank-metric codes with given ranks (GFRMCs), to generalize the parallel construction and the parallel multilevel construction in [IEEE Trans. Inf. Theory, 66 (2020), 6884–6897]. The lower bounds for GFRMCs are derived from Ferrers diagram rank-metric codes (FDRMCs). Via GFRMCs, the inverse multilevel construction for CDCs is showed. Furthermore, the double multilevel construction, as an effective construction for CDCs, is presented by combining the inverse multilevel construction and the multilevel construction. Many CDCs with larger size than the previously best known codes are given.

The interplay of different metrics for the construction of constant dimension codes

&lt;p style='text-indent:20px;'&gt;A basic problem for constant dimension codes is to determine the maximum possible size &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ A_q(n,d;k) $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; of a set of &lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$ k $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;-dimensional subspaces in &lt;inline-formula&gt;&lt;tex-math id="M3"&gt;\begin{document}$ \mathbb{F}_q^n $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;, called codewords, such that the subspace distance satisfies &lt;inline-formula&gt;&lt;tex-math id="M4"&gt;\begin{document}$ d_S(U,W): = 2k-2\dim(U\cap W)\ge d $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; for all pairs of different codewords &lt;inline-formula&gt;&lt;tex-math id="M5"&gt;\begin{document}$ U $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;, &lt;inline-formula&gt;&lt;tex-math id="M6"&gt;\begin{document}$ W $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;. Constant dimension codes have applications in e.g. random linear network coding, cryptography, and distributed storage. Bounds for &lt;inline-formula&gt;&lt;tex-math id="M7"&gt;\begin{document}$ A_q(n,d;k) $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; are the topic of many recent research papers. Providing a general framework we survey many of the latest constructions and show the potential for further improvements. As examples we give improved constructions for the cases &lt;inline-formula&gt;&lt;tex-math id="M8"&gt;\begin{document}$ A_q(10,4;5) $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;, &lt;inline-formula&gt;&lt;tex-math id="M9"&gt;\begin{document}$ A_q(11,4;4) $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;, &lt;inline-formula&gt;&lt;tex-math id="M10"&gt;\begin{document}$ A_q(12,6;6) $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;, and &lt;inline-formula&gt;&lt;tex-math id="M11"&gt;\begin{document}$ A_q(15,4;4) $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;. We also derive general upper bounds for subcodes arising in those constructions.&lt;/p&gt;

Improved generalized block inserting construction of constant dimension codes

Improved generalized block inserting construction of constant dimension codes

Generalized block inserting for constructing new constant dimension codes

Generalized block inserting for constructing new constant dimension codes

A stability result and a spectrum result on constant dimension codes

In this paper we continue the study of subspace codes with constant intersection dimension (SCIDs). We investigate the largest possible dimensions spanned by non-sunflower SCIDs, by proving a stability result and we present a spectrum result giving us a large interval for the dimensions that can be generated by non-sunflower SCIDs. We also prove that SCIDs are in fact equivalent to linear spaces and give some results regarding this equivalence, including an improvement to a result by E. Gorla and A. Ravagnani, regarding the number of centers of a SCID.

Constructions of Orbit Codes Based on Unitary Spaces Over Finite Fields

Orbit codes, as special constant dimension codes, have attracted much attention due to their applications for error correction in random network coding. This paper is devoted to constructing large orbit codes by making full use of unitary space. Firstly, we construct a cyclic unitary group of order $q^{2n}-1$ by means of the companion matrix of a primitive polynomial over finite fields $\mathbb {F}_{q^{2}}$ , and so the corresponding code is unitary cyclic orbit code. As a special application, a new quaternary orbit code $(6,63,4,3)$ is given. Secondly, we obtain orbit codes with large size using the external direct product of unitary groups acting on the direct sum of subspaces. Finally, a table is given for illustrating our codes improve upon those constructed by Trautmann et al. and Poroch et al.

New Constant Dimension Subspace Codes From Generalized Inserting Construction

A basic problem about a constant dimension subspace code is to find its maximal possible size A <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> (n, d, k). Linkage construction and inserting construction are two effective methods to improve lower bounds for constant dimension codes. This letter concentrates on parallel linkage construction by He and direct inserting construction by Lao and generalizes the previous results to obtain new lower bounds of constant dimension codes, such as Aq(12,4,6).

Generalized Linkage Construction for Constant-Dimension Codes

A constant-dimension code (CDC) is a set of subspaces of constant dimension in a common vector space with upper bounded pairwise intersection. We improve and generalize two constructions for CDCs, the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">improved linkage construction</i> and the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">parallel linkage construction</i> , to the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">generalized linkage construction</i> and the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">multiblock generalized linkage construction</i> which in turn yield many improved lower bounds for the cardinalities of CDCs; a quantity not known in general.

Generalized LMRD Code Bounds for Constant Dimension Codes

In random network coding so-called constant dimension codes (CDCs) are used for error correction and detection. Most of the largest known codes contain a lifted maximum rank distance (LMRD) code as a subset. For some special cases, Etzion and Silberstein have demonstrated that one can obtain tighter upper bounds on the maximum possible cardinality of CDCs if we assume that an LMRD code is contained. The range of applicable parameters was partially extended by Heinlein. Here we fully generalize those bounds, which also sheds some light on recent constructions.

Improving the Linkage Construction With Echelon-Ferrers for Constant-Dimension Codes

Echelon-Ferrers is an important method to improve lower bounds for constant-dimension codes, which can be applied on various parameters. Fagang Li combined the linkage construction and echelon-Ferrers to obtain some new lower bounds of constant-dimension codes. In this letter, we generalize this linkage construction to obtain new lower bounds.

Parallel Multilevel Constructions for Constant Dimension Codes

Constant dimension codes (CDCs), as special subspace codes, have received a lot of attention due to their application in random network coding. This paper introduces a family of new codes, called rank metric codes with given ranks (GRMCs), to generalize the parallel construction in [Xu and Chen, IEEE Trans. Inf. Theory, 64 (2018), 6315–6319] and the classic multilevel construction. A Singleton-like upper bound and a lower bound for GRMCs derived from Gabidulin codes are given. Via GRMCs, two effective constructions for CDCs are presented by combining the parallel construction and the multilevel construction. Many CDCs with larger size than the previously best known codes are given. The ratio between the new lower bound and the known upper bound for $(4\delta,2\delta,2\delta)_{q}$ -CDCs is calculated. It is greater than 0.99926 for any prime power $q$ and any $\delta \geq 3$ .

No Projective 16-Divisible Binary Linear Code of Length 131 Exists

We show that no projective 16-divisible binary linear code of length 131 exists. This implies several improved upper bounds for constant-dimension codes, used in random linear network coding, and partial spreads.

Deterministic construction of compressed sensing matrices from constant dimension codes

Compressed sensing theory provides a new approach to acquire data as a sampling technique and makes sure that sparse signals can be reconstructed from few measurements. The construction of compressed sensing matrices is a central problem in compressed sensing theory. In this paper, the deterministic sparse compressed sensing matrices from constant dimension codes are constructed and the coherence of the matrices are computed. Furthermore, the maximum sparsity of recovering the sparse signals by using our matrices is obtained. Meanwhile, the numerical simulations are made among our matrices from constant dimension codes, R. DeVore’s matrices, matrices from p-ary BCH codes and random matrices. Moreover, the deterministic sparse compressed sensing matrices can be constructed from certain Steiner structures. Finally, a general analysis indicates that the deterministic sparse compressed sensing matrices from constant dimension codes can recover signals of arbitrary sparsity order with any number of budged rows in some Euclid spaces with proper dimensions.