scholarly journals Bounds on Subspace Codes Based on Subspaces of Type(m,1)in Singular Linear Space

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
You Gao ◽  
Gang Wang
Keyword(s):  
Linear Space ◽  
Finite Fields ◽  
Sphere Packing ◽  
Subspace Codes ◽  
Singleton Bound ◽  
Johnson Bound

The Sphere-packing bound, Singleton bound, Wang-Xing-Safavi-Naini bound, Johnson bound, and Gilbert-Varshamov bound on the subspace codesn+l,M,d,(m,1)qbased on subspaces of type(m,1)in singular linear spaceFq(n+l)over finite fieldsFqare presented. Then, we prove that codes based on subspaces of type(m,1)in singular linear space attain the Wang-Xing-Safavi-Naini bound if and only if they are certain Steiner structures inFq(n+l).

Bounds on subspace codes based on totally isotropic subspaces in unitary spaces

2016 ◽  
Vol 08 (04) ◽  
pp. 1650056 ◽  
Author(s):  
You Gao ◽  
Liyum Zhao ◽  
Gang Wang
Keyword(s):  
Finite Fields ◽  
Sphere Packing ◽  
Unitary Space ◽  
Subspace Codes ◽  
Singleton Bound ◽  
Johnson Bound ◽  
Isotropic Subspaces ◽  
Totally Isotropic Subspaces

In this paper, the Sphere-packing bound, Singleton bound, Wang–Xing–Safavi-Naini bound, Johnson bound and Gilbert–Varshamov bound on the subspace codes [Formula: see text] based on [Formula: see text]-dimensional totally isotropic subspaces in unitary space [Formula: see text] over finite fields [Formula: see text] are presented. Then, we prove that [Formula: see text] codes based on [Formula: see text]-dimensional totally isotropic subspaces in unitary space [Formula: see text] attain the Wang–Xing–Safavi-Naini bound if and only if they are certain Steiner structures in [Formula: see text].

Bounds on Subspace Codes Based on Orthogonal Space Over Finite Fields of Characteristic 2

2019 ◽  
Vol 30 (05) ◽  
pp. 735-757
Author(s):  
Gang Wang ◽  
Min-Yao Niu ◽  
Fang-Wei Fu
Keyword(s):  
Linear Programming ◽  
Positive Integer ◽  
Finite Fields ◽  
Sphere Packing ◽  
Singular Subspace ◽  
Subspace Codes ◽  
Orthogonal Space ◽  
Johnson Bound ◽  
Characteristic 2 ◽  
Subspace Distance

In this paper, the Sphere-packing bound, Wang-Xing-Safavi-Naini bound, Johnson bound and Gilbert-Varshamov bound on the subspace code of length [Formula: see text], size [Formula: see text], minimum subspace distance [Formula: see text] based on [Formula: see text]-dimensional totally singular subspace in the [Formula: see text]-dimensional orthogonal space [Formula: see text] over finite fields [Formula: see text] of characteristic 2, denoted by [Formula: see text], are presented, where [Formula: see text] is a positive integer, [Formula: see text], [Formula: see text], [Formula: see text]. Then, we prove that [Formula: see text] codes attain the Wang-Xing-Safavi-Naini bound if and only if they are certain Steiner structures in [Formula: see text], where [Formula: see text] denotes the collection of all the [Formula: see text]-dimensional totally singular subspaces in the [Formula: see text]-dimensional orthogonal space [Formula: see text] over [Formula: see text] of characteristic 2. Finally, Gilbert-Varshamov bound and linear programming bound on the subspace code [Formula: see text] in [Formula: see text] are provided, where [Formula: see text] denotes the collection of all the totally singular subspaces in the [Formula: see text]-dimensional orthogonal space [Formula: see text] over [Formula: see text] of characteristic 2.

Bounds on subspace codes based on totally isotropic subspace in symplectic spaces and extended symplectic spaces

2019 ◽  
Vol 12 (05) ◽  
pp. 1950069
Author(s):  
Mahdieh Hakimi Poroch
Keyword(s):  
Sphere Packing ◽  
Symplectic Space ◽  
Isotropic Subspace ◽  
Subspace Codes ◽  
Singleton Bound ◽  
Symplectic Spaces ◽  
Isotropic Subspaces ◽  
Totally Isotropic Subspace ◽  
Totally Isotropic Subspaces

In this paper, we propose the Sphere-packing bound, Singleton bound and Gilbert–Varshamov bound on the subspace codes [Formula: see text] based on totally isotropic subspaces in symplectic space [Formula: see text] and on the subspace codes [Formula: see text] based on totally isotropic subspace in extended symplectic space [Formula: see text].

scholarly journals Subspace Codes Based on Partial Injective Maps of Vector Spaces Over Finite Fields

IEEE Access ◽  
2020 ◽  
Vol 8 ◽  
pp. 192608-192615
Author(s):  
Hong-Li Wang ◽  
Gang Wang ◽  
You Gao
Keyword(s):  
Finite Fields ◽  
Vector Spaces ◽  
Subspace Codes ◽  
Injective Maps

scholarly journals Johnson Type Bounds for Mixed Dimension Subspace Codes

10.37236/8188 ◽  
2019 ◽  
Vol 26 (3) ◽  
Author(s):  
Thomas Honold ◽  
Michael Kiermaier ◽  
Sascha Kurz
Keyword(s):  
Network Coding ◽  
Upper Bounds ◽  
Linear Network ◽  
Subspace Codes ◽  
Linear Network Coding ◽  
Constant Dimension ◽  
Random Linear Network Coding ◽  
Johnson Bound ◽  
Constant Dimension Codes ◽  
Mixed Dimension

Subspace codes, i.e., sets of subspaces of $\mathbb{F}_q^v$, are applied in random linear network coding. Here we give improved upper bounds for their cardinalities based on the Johnson bound for constant dimension codes.

scholarly journals New Construction of Maximum Distance Separable (MDS) Self-Dual Codes over Finite Fields

Entropy ◽  
2019 ◽  
Vol 21 (2) ◽  
pp. 101 ◽  
Author(s):  
Aixian Zhang ◽  
Zhe Ji
Keyword(s):  
Finite Fields ◽  
Maximum Distance ◽  
Dual Codes ◽  
Self Duality ◽  
Reed Solomon Codes ◽  
Interesting Topic ◽  
Singleton Bound ◽  
Maximum Distance Separable ◽  
New Construction

Maximum distance separable (MDS) self-dual codes have useful properties due to their optimality with respect to the Singleton bound and its self-duality. MDS self-dual codes are completely determined by the length n , so the problem of constructing q-ary MDS self-dual codes with various lengths is a very interesting topic. Recently X. Fang et al. using a method given in previous research, where several classes of new MDS self-dual codes were constructed through (extended) generalized Reed-Solomon codes, in this paper, based on the method given in we achieve several classes of MDS self-dual codes.

scholarly journals Construction of random pooling designs based on singular linear space over finite fields

2021 ◽  
Vol 7 (3) ◽  
pp. 4376-4385
Author(s):  
Xuemei Liu ◽  
◽  
Yazhuo Yu
Keyword(s):  
Linear Space ◽  
Finite Fields ◽  
Pooling Designs ◽  
Linear Spaces ◽  
Disjunct Matrix ◽  
Short Time ◽  
Singular Linear Spaces

<abstract><p>Faced with a large number of samples to be tested, if there are requiring to be tested one by one and complete in a short time, it is difficult to save time and save costs at the same time. The random pooling designs can deal with it to some degree. In this paper, a family of random pooling designs based on the singular linear spaces and related counting theorems are constructed. Furtherly, based on it we construct an $ \alpha $-$ almost\ d^e $-disjunct matrix and an $ \alpha $-$ almost\ (d, r, z] $-disjunct matrix, and all the parameters and properties of these random pooling designs are given. At last, by comparing to Li's construction, we find that our design is better under certain condition.</p></abstract>

scholarly journals NETWORK CODING WITH MODULAR LATTICES

2011 ◽  
Vol 10 (06) ◽  
pp. 1319-1342 ◽  
Author(s):  
ANDREAS KENDZIORRA ◽  
STEFAN E. SCHMIDT
Keyword(s):  
Vector Space ◽  
Network Coding ◽  
Sphere Packing ◽  
Height Function ◽  
Error Correcting Codes ◽  
Modular Lattices ◽  
Subspace Lattice ◽  
New Model ◽  
Singleton Bound ◽  
Sphere Covering

Kötter and Kschischang presented in 2008 a new model for error correcting codes in network coding. The alphabet in this model is the subspace lattice of a given vector space, a code is a subset of this lattice and the used metric on this alphabet is the map d : (U, V) ↦ dim (U+V)- dim (U∩V). In this paper we generalize this model to arbitrary modular lattices, i.e. we consider codes, which are subsets of modular lattices. The used metric in this general case is the map d : (u, v) ↦ h(u ∨ v) - h(u ∧ v), where h is the height function of the lattice. We apply this model to submodule lattices. Moreover, we show a method to compute the size of spheres in certain modular lattices and present a sphere packing bound, a sphere covering bound, and a Singleton bound for codes, which are subsets of modular lattices.

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