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].

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].

scholarly journals Calculating the Dimension of the Universal Embedding of the Symplectic Dual Polar Space using Languages

10.37236/9754 ◽  
2020 ◽  
Vol 27 (4) ◽  
Author(s):  
Carlos Segovia ◽  
Monika Winklmeier
Keyword(s):  
Polar Space ◽  
Symplectic Space ◽  
Alternative Proof ◽  
Isotropic Subspace ◽  
Dual Polar Space ◽  
Standard Order ◽  
Isotropic Subspaces ◽  
Universal Embedding ◽  
Totally Isotropic Subspaces ◽  
Symplectic Dual Polar Space

The main result of this paper is the construction of a bijection of the set of words in so-called standard order of length $n$ formed by four different letters and the set $\mathcal{N}^n$ of all subspaces of a fixed $n$-dimensional maximal isotropic subspace of the $2n$-dimensional symplectic space $V$ over $\mathbb{F}_2$ which are not maximal in a certain sense. Since the number of different words in standard order is known, this gives an alternative proof for the formula of the dimension of the universal embedding of a symplectic dual polar space $\mathcal{G}_n$. Along the way, we give formulas for the number of all $n$- and $(n-1)$-dimensional totally isotropic subspaces of $V$.

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).

Constructions of (r,t)-LRC Based on Totally Isotropic Subspaces in Symplectic Space Over Finite Fields

2020 ◽  
Vol 31 (03) ◽  
pp. 327-339
Author(s):  
Gang Wang ◽  
Min-Yao Niu ◽  
Fang-Wei Fu
Keyword(s):  
Finite Fields ◽  
Linear Codes ◽  
Distributed Storage ◽  
Symplectic Space ◽  
Information Rate ◽  
Local Repair ◽  
Distributed Storage Systems ◽  
Isotropic Subspaces ◽  
Locally Repairable Codes ◽  
Totally Isotropic Subspaces

Linear code with locality [Formula: see text] and availability [Formula: see text] is that the value at each coordinate [Formula: see text] can be recovered from [Formula: see text] disjoint repairable sets each containing at most [Formula: see text] other coordinates. This property is particularly useful for codes in distributed storage systems because it permits local repair of failed nodes and parallel access of hot data. In this paper, two constructions of [Formula: see text]-locally repairable linear codes based on totally isotropic subspaces in symplectic space [Formula: see text] over finite fields [Formula: see text] are presented. Meanwhile, comparisons are made among the [Formula: see text]-locally repairable codes we construct, the direct product code in Refs. [8], [11] and the codes in Ref. [9] about the information rate [Formula: see text] and relative distance [Formula: see text].

scholarly journals Totally isotropic subspaces of small height in quadratic spaces

2016 ◽  
Vol 16 (2) ◽  
Author(s):  
Wai Kiu Chan ◽  
Lenny Fukshansky ◽  
Glenn R. Henshaw
Keyword(s):  
Global Field ◽  
Small Height ◽  
Isotropic Subspaces ◽  
Totally Isotropic Subspaces

AbstractLet K be a global field or

On the Pauli graphs on N-qudits

2008 ◽  
Vol 8 (1&2) ◽  
pp. 127-146
Author(s):  
M. Planat ◽  
M. Saniga
Keyword(s):  
Independent Set ◽  
Generalized Quadrangle ◽  
Maximum Independent Set ◽  
Partial Geometry ◽  
Commuting Operators ◽  
Strongly Regular ◽  
Isotropic Subspaces ◽  
Commuting Pairs ◽  
Set Of Points ◽  
Totally Isotropic Subspaces

A comprehensive graph theoretical and finite geometrical study of the commutation relations between the generalized Pauli operators of $N$-qudits is performed in which vertices/points correspond to the operators and edges/lines join commuting pairs of them. As per two-qubits, all basic properties and partitionings of the corresponding {\it Pauli graph} are embodied in the geometry of the generalized quadrangle of order two. Here, one identifies the operators with the points of the quadrangle and groups of maximally commuting subsets of the operators with the lines of the quadrangle. The three basic partitionings are (a) a pencil of lines and a cube, (b) a Mermin's array and a bipartite-part and (c) a maximum independent set and the Petersen graph. These factorizations stem naturally from the existence of three distinct geometric hyperplanes of the quadrangle, namely a set of points collinear with a given point, a grid and an ovoid, which answer to three distinguished subsets of the Pauli graph, namely a set of six operators commuting with a given one, a Mermin's square, and set of five mutually non-commuting operators, respectively. The generalized Pauli graph for multiple qubits is found to follow from symplectic polar spaces of order two, where maximal totally isotropic subspaces stand for maximal subsets of mutually commuting operators. The substructure of the (strongly regular) $N$-qubit Pauli graph is shown to be pseudo-geometric, i.\,e., isomorphic to a graph of a partial geometry. Finally, the (not strongly regular) Pauli graph of a two-qutrit system is introduced; here it turns out more convenient to deal with its dual in order to see all the parallels with the two-qubit case and its surmised relation with the generalized quadrangle $Q(4,3)$, the dual of $W(3)$.

scholarly journals SMALL ZEROS OF QUADRATIC FORMS OVER $\overline{\mathbb{Q}}$

2008 ◽  
Vol 04 (03) ◽  
pp. 503-523 ◽  
Author(s):  
LENNY FUKSHANSKY
Keyword(s):  
Quadratic Form ◽  
Quadratic Forms ◽  
Number Fields ◽  
Dimensional Subspace ◽  
Orthogonal Decomposition ◽  
Isotropic Subspace ◽  
Effective Version ◽  
Small Height ◽  
Totally Isotropic Subspace

Let N ≥ 2 be an integer, F a quadratic form in N variables over [Formula: see text], and [Formula: see text] an L-dimensional subspace, 1 ≤ L ≤ N. We prove the existence of a small-height maximal totally isotropic subspace of the bilinear space (Z,F). This provides an analogue over [Formula: see text] of a well-known theorem of Vaaler proved over number fields. We use our result to prove an effective version of Witt decomposition for a bilinear space over [Formula: see text]. We also include some related effective results on orthogonal decomposition and structure of isometries for a bilinear space over [Formula: see text]. This extends previous results of the author over number fields. All bounds on height are explicit.

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