Enumerative Coding for Grassmannian Space

The Grassmannian space G q (n, k) is the set of all k-dimensional subspaces of the vector space F q n . Recently, codes in the Grassmannian have found an application in network coding. The main goal of this paper is to present efficient enumerative encoding and decoding techniques for the Grassmanni...

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Veröffentlicht in:IEEE transactions on information theory Jg. 57; H. 1; S. 365 - 374
Hauptverfasser: Silberstein, N, Etzion, T
Format: Journal Article
Sprache:Englisch
Veröffentlicht: New York, NY IEEE 01.01.2011
Institute of Electrical and Electronics Engineers
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Schlagworte:
Applied sciences
Codes
Coding
Coding, codes
Complex systems
Complexity
Complexity theory
Decoding
Digits
Encoding
Enumerative coding
Error correction codes
Exact sciences and technology
Ferrers diagram
Graphics
Grassmannian
identifying vector
Indexes
Information theory
Information, signal and communications theory
Mathematical analysis
Methods
Network coding
partitions
reduced row echelon form
Representations
Signal and communications theory
Subspaces
Telecommunications and information theory
Vector space
Vectors (mathematics)
Enumerative coding
partitions
Ferrers diagram
Vector space
Coding
Subspace method
Grassmannian
Decoding
reduced row echelon form
identifying vector
ISSN:0018-9448, 1557-9654
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Beschreibung
Zusammenfassung:The Grassmannian space G q (n, k) is the set of all k-dimensional subspaces of the vector space F q n . Recently, codes in the Grassmannian have found an application in network coding. The main goal of this paper is to present efficient enumerative encoding and decoding techniques for the Grassmannian. These coding techniques are based on two different orders for the Grassmannian induced by different representations of k-dimensional subspaces of F q n . One enumerative coding method is based on a Ferrers diagram representation and on an order for G q (n, k) based on this representation. The complexity of this enumerative coding is O(k 5/2 (n - k) 5/2 ) digit operations. Another order of the Grassmannian is based on a combination of an identifying vector and a reduced row echelon form representation of subspaces. The complexity of the enumerative coding, based on this order, is O(nk(n - k) log n log log n) digit operations. A combination of the two methods reduces the complexity on average by a constant factor.
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ISSN:0018-9448
1557-9654
DOI:10.1109/TIT.2010.2090252