Improved Expression for Rank Distribution of Sparse Random Linear Network Coding

Characterization of the rank distribution of a sparse random matrix over a finite field can be decomposed into two subproblems. The first subproblem is the characterization of the probability <inline-formula> <tex-math notation="LaTeX">p(i,n) </tex-math></inline-formul...

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Bibliographic Details
Published in:IEEE communications letters Vol. 25; no. 5; pp. 1472 - 1476
Main Authors: Chen, Wenlin, Lu, Fang, Dong, Yan
Format: Journal Article
Language:English
Published: New York IEEE 01.05.2021
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subjects:
Decoding
Decomposition
Eigenvalues and eigenfunctions
Exact solutions
Fields (mathematics)
Markov chains
Markov processes
Matrices (mathematics)
Matrix decomposition
Network coding
rank distribution
Receivers
Sparse matrices
Sparse random linear network coding
ISSN:1089-7798, 1558-2558
Online Access:Get full text
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Summary:Characterization of the rank distribution of a sparse random matrix over a finite field can be decomposed into two subproblems. The first subproblem is the characterization of the probability <inline-formula> <tex-math notation="LaTeX">p(i,n) </tex-math></inline-formula> that an <inline-formula> <tex-math notation="LaTeX">n </tex-math></inline-formula>-dimensional vector is linearly dependent of other <inline-formula> <tex-math notation="LaTeX">i </tex-math></inline-formula> linearly independent <inline-formula> <tex-math notation="LaTeX">n </tex-math></inline-formula>-dimensional vectors. The second subproblem is the characterization of the rank distribution of a sparse random matrix as a function of <inline-formula> <tex-math notation="LaTeX">p(i,n) </tex-math></inline-formula>. In this letter, we focus on the second subproblem and present an exact solution to it. The derivation is based on an absorbing Markov chain and the eigen decomposition of the transition matrix. Compared with the state-of-the-art expressions, the derived expression is closed-form and of lower complexity. As a necessity for the exactness of the derived expression, we prove that the derived expression is equivalent to the state-of-the-art expressions.
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ISSN:1089-7798
1558-2558
DOI:10.1109/LCOMM.2020.3041845