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The rank of sparse random matrices over finite fields

1997
*
Random structures & algorithms (Print)
*

depending on n and is unboundedasn goes to in nity then

doi:10.1002/(sici)1098-2418(199707)10:4<407::aid-rsa1>3.0.co;2-y
fatcat:ybnhml52ujaz5iwfvsgsc4vu5a
*the*expected di erence between*the**rank**of*M and n is unbounded. ... It is shown that*the*expected*rank**of*M is n ; O (1): Furthermore, there is a constant A such that*the*probability that*the**rank*is less than n ; k is less than A=q k : It is also shown that if c grows ... Introduction In this paper we i n vestigate*the**rank**of**random**matrices**over*a xed but arbitrary nite eld GF q]. ...##
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Sparse Recovery Using Sparse Sensing Matrix Based Finite Field Optimization in Network Coding

2017
*
IEICE transactions on information and systems
*

In this paper, we propose a

doi:10.1587/transinf.2016edl8189
fatcat:l22l7immazeztdwiqjg32gquca
*sparse*recovery approach using*sparse*sensing matrix to solve*the*NC all-or-nothing problem*over*a*finite**field*. ...*The*effectiveness*of**the*proposed approach is evaluated based on a sensor network. ... Conclusion We have proposed a new framework for*sparse*recovery*over*a*finite**field*with*sparse**random*network transfer*matrices*that can overcome*the*all-or-nothing problem in NC. ...##
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On the Compressed Measurements over Finite Fields: Sparse or Dense Sampling
[article]

2012
*
arXiv
*
pre-print

Our results are obtained while

arXiv:1211.5207v1
fatcat:7c6cipsm6bca5pf7beix3f5nl4
*the**sparseness**of**the*sensing*matrices*as well as*the*size*of**the**finite**fields*are varied. ... We consider compressed sampling*over**finite**fields*and investigate*the*number*of*compressed measurements needed for successful L0 recovery. ... There are a couple*of*related works. Draper and Malekpour [3] reported on*the*error exponents for recovery*of**sparse*signals using uniform*random*sensing*matrices**over**finite**fields*. ...##
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Rank minimization over finite fields

2011
*
2011 IEEE International Symposium on Information Theory Proceedings
*

This paper establishes information-theoretic limits in estimating a

doi:10.1109/isit.2011.6033722
dblp:conf/isit/TanBD11
fatcat:gcstzavxgrd5ja4iwhuv44sufm
*finite**field*low-*rank*matrix given*random*linear measurements*of*it. ...*The*reliability function associated to*the*minimum-*rank*decoder is also derived. Our bounds hold even in*the*case where*the*sensing*matrices*are*sparse*. Connections to*rank*-metric codes are discussed. ...*The*family*of*codes known as*rank*-metric codes [6] - [8] is similar to*the**rank*minimization problem*over**finite**fields*. We comment on connections in Section VI and [12] . II. ...##
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Rank Minimization Over Finite Fields: Fundamental Limits and Coding-Theoretic Interpretations

2012
*
IEEE Transactions on Information Theory
*

This paper establishes information-theoretic limits in estimating a

doi:10.1109/tit.2011.2178017
fatcat:buijac3fana2louxc4u2rlwgym
*finite**field*low-*rank*matrix given*random*linear measurements*of*it. ... These linear measurements are obtained by taking inner products*of**the*low-*rank*matrix with*random*sensing*matrices*. ... Acknowledgements*The*authors would like to thank Ying Liu (MIT) for his detailed comments.*The*authors would also like to thank Huili Guo (MIT) for her help in generating Fig. 2 . ...##
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LINBOX: A GENERIC LIBRARY FOR EXACT LINEAR ALGEBRA

2002
*
Mathematical Software
*

Preconditioning for

doi:10.1142/9789812777171_0005
fatcat:z7vijbo36ffh7hfbgd6epjmlam
*the**rank*and*the*determinant*over**finite**fields*. ... Minimal polynomial and linear system solution*over**finite**fields*. ...##
###
The Decoding Success Probability of Sparse Random Linear Network Coding for Multicast
[article]

2020
*
arXiv
*
pre-print

A fundamental problem

arXiv:2010.05555v1
fatcat:ubdv45ck25flrogrc5vfnfkkxu
*of*such communication is to characterize*the*decoding success probability, which is given by*the*probability*of*a*sparse**random*matrix*over*a*finite**field*being full*rank*. ... In this paper, we provide a tight and closed-form approximation to*the*probability*of*a*sparse**random*matrix being full*rank*, by presenting*the*explicit structure*of**the*reduced row echelon form*of*a full ... Then we established an exact expression for*the**rank*distribution*of**sparse**random**matrices**over*as a function*of*( , ). ...##
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High-Girth Matrices and Polarization
[article]

2015
*
arXiv
*
pre-print

*Random*

*matrices*can be used to show

*the*existence

*of*high-girth

*matrices*with constant relative

*rank*, but

*the*construction is non-explicit. ... This paper uses a polar-like construction to obtain a deterministic and efficient construction

*of*high-girth

*matrices*for arbitrary

*fields*and relative

*ranks*. ... ACKNOWLEDGEMENTS This work was partially supported by NSF grant CIF-1706648 and

*the*Princeton PACM Director's Fellowship. ...

##
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Faster Sparse Matrix Inversion and Rank Computation in Finite Fields
[article]

2021
*
arXiv
*
pre-print

We achieve

arXiv:2106.09830v1
fatcat:2q6qezj2nrdkbmouufmhm53khy
*the*same running time for*the*computation*of**the**rank*and nullspace*of*a*sparse*matrix*over*a*finite**field*. This improvement relies on two key techniques. ... We improve*the*current best running time value to invert*sparse**matrices**over**finite**fields*, lowering it to an expected O(n^2.2131) time for*the*current values*of*fast rectangular matrix multiplication ... In this paper, we study*the*problem*of*matrix inversion and*rank*computation*of*an n × n matrix A*over*a*finite**field*, focusing on*sparse**matrices*and certain other classes*of*structured*matrices*. ...##
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Computational linear algebra over finite fields
[article]

2012
*
arXiv
*
pre-print

We present here algorithms for efficient computation

arXiv:1204.3735v1
fatcat:a3j26roivfd55fsf3sk7ahxgk4
*of*linear algebra problems*over**finite**fields*. ... Remark 25*Over*small*fields*, if*the**rank**of**the*matrix is known,*the*diagonal*matrices**of*line 1 can be replaced by*sparse*preconditioners with O (n log(n)) nonzero coefficients to avoid*the*need*of**field*... ij = a ij + δ i * a kj ), as is*the*case with*matrices**over**finite**fields*...##
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Efficient matrix preconditioners for black box linear algebra

2002
*
Linear Algebra and its Applications
*

*The*focus is on linear algebra problems

*over*

*finite*

*fields*, but most results are valid for entries from arbitrary

*fields*. ... We present new conditioners, including conditioners to preserve low displacement

*rank*for Toeplitz-like

*matrices*. ... Acknowledgements This material is based on

*the*work supported in part by

*the*National Science ...

##
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Parallel computation of the rank of large sparse matrices from algebraic K-theory
[article]

2007
*
arXiv
*
pre-print

This paper deals with

arXiv:0704.2351v2
fatcat:rbmrycton5g7fkgee4lfxjb7ve
*the*computation*of**the**rank*and*of*some integer Smith forms*of*a series*of**sparse**matrices*arising in algebraic K-theory. ...*The*number*of*non zero entries in*the*considered*matrices*ranges from 8 to 37 millions.*The*largest*rank*computation took more than 35 days on 50 processors. ... Thus, by using well chosen preconditioners and Wiedemann algorithm one can easily compute*the**rank**of*a*sparse*matrix*over*a*finite**field*. ...##
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Parallel computation of the rank of large sparse matrices from algebraic K-theory

2007
*
Proceedings of the 2007 international workshop on Parallel symbolic computation - PASCO '07
*

This paper deals with

doi:10.1145/1278177.1278186
dblp:conf/issac/DumasEGU07
fatcat:tqci4jhlfnbv3eptehnioonq5i
*the*computation*of**the**rank*and some integer Smith forms*of*a series*of**sparse**matrices*arising in algebraic K-theory. ...*The*number*of*non zero entries in*the*considered*matrices*ranges from 8 to 37 millions.*The*largest*rank*computation took more than 35 days on 50 processors. ... Thus, by using well chosen preconditioners and Wiedemann algorithm one can easily compute*the**rank**of*a*sparse*matrix*over*a*finite**field*. ...##
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Stein's method and the rank distribution of random matrices over finite fields

2015
*
Annals of Probability
*

With Q_q,n

doi:10.1214/13-aop889
fatcat:7tnyo3dnjzedfhetqruklhisem
*the*distribution*of*n minus*the**rank**of*a matrix chosen uniformly from*the*collection*of*all n×(n+m)*matrices**over**the**finite**field*F_q*of*size q>2, and Q_q*the*distributional limit*of*Q_q,n ... In addition, we obtain similar sharp results for*the**rank*distributions*of*symmetric, symmetric with zero diagonal, skew symmetric, skew centrosymmetric and Hermitian*matrices*. ...*The*next paragraph gives pointers to*the*large literature on*ranks**of**random**matrices**over**finite**fields*. ...##
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Multiple Target Localization in WSNs Based on Compressive Sensing Using Deterministic Sensing Matrices

2015
*
International Journal of Distributed Sensor Networks
*

Further simulation shows that

doi:10.1155/2015/947016
fatcat:5xaxifvpnbc5dony6k4knpuzim
*the*proposed approach is practical in use, while being favorably comparable to*the*existing*random*sensing*matrices*in reconstruction performance. ... For this purpose,*random*sensing*matrices*have been studied, while a few researches on deterministic sensing*matrices*have been considered. ... In [23] , a new method to construct binary sensing*matrices*is introduced by using algebraic curves*over*a*finite**field*. ...
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