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The rank of sparse random matrices over finite fields

Johannes Bl�mer, Richard Karp, Emo Welzl
1997 Random structures & algorithms (Print)  
depending on n and is unboundedasn goes to in nity then 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].  ... 
doi:10.1002/(sici)1098-2418(199707)10:4<407::aid-rsa1>3.0.co;2-y fatcat:ybnhml52ujaz5iwfvsgsc4vu5a

Sparse Recovery Using Sparse Sensing Matrix Based Finite Field Optimization in Network Coding

Ganzorig GANKHUYAG, Eungi HONG, Yoonsik CHOE
2017 IEICE transactions on information and systems  
In this paper, we propose a 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.  ... 
doi:10.1587/transinf.2016edl8189 fatcat:l22l7immazeztdwiqjg32gquca

On the Compressed Measurements over Finite Fields: Sparse or Dense Sampling [article]

Jin-Taek Seong, Heung-No Lee
2012 arXiv   pre-print
Our results are obtained while 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.  ... 
arXiv:1211.5207v1 fatcat:7c6cipsm6bca5pf7beix3f5nl4

Rank minimization over finite fields

Vincent Y. F. Tan, Laura Balzano, Stark C. Draper
2011 2011 IEEE International Symposium on Information Theory Proceedings  
This paper establishes information-theoretic limits in estimating a 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.  ... 
doi:10.1109/isit.2011.6033722 dblp:conf/isit/TanBD11 fatcat:gcstzavxgrd5ja4iwhuv44sufm

Rank Minimization Over Finite Fields: Fundamental Limits and Coding-Theoretic Interpretations

V. Y. F. Tan, L. Balzano, S. C. Draper
2012 IEEE Transactions on Information Theory  
This paper establishes information-theoretic limits in estimating a 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 .  ... 
doi:10.1109/tit.2011.2178017 fatcat:buijac3fana2louxc4u2rlwgym

LINBOX: A GENERIC LIBRARY FOR EXACT LINEAR ALGEBRA

J.-G. DUMAS, T. GAUTIER, M. GIESBRECHT, P. GIORGI, B. HOVINEN, E. KALTOFEN, B. D. SAUNDERS, W. J. TURNER, G. VILLARD
2002 Mathematical Software  
Preconditioning for the rank and the determinant over finite fields.  ...  Minimal polynomial and linear system solution over finite fields.  ... 
doi:10.1142/9789812777171_0005 fatcat:z7vijbo36ffh7hfbgd6epjmlam

The Decoding Success Probability of Sparse Random Linear Network Coding for Multicast [article]

Wenlin Chen, Fang Lu, Yan Dong
2020 arXiv   pre-print
A fundamental problem 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 ( , ).  ... 
arXiv:2010.05555v1 fatcat:ubdv45ck25flrogrc5vfnfkkxu

High-Girth Matrices and Polarization [article]

Emmanuel Abbe, Yuval Wigderson
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.  ... 
arXiv:1501.06528v2 fatcat:gk3ai2y5h5cezcnfyzaiyqzoy4

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

Sílvia Casacuberta, Rasmus Kyng
2021 arXiv   pre-print
We achieve 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.  ... 
arXiv:2106.09830v1 fatcat:2q6qezj2nrdkbmouufmhm53khy

Computational linear algebra over finite fields [article]

Jean-Guillaume Dumas, Clément Pernet (INRIA Grenoble Rhône-Alpes / LIG Laboratoire d'Informatique de Grenoble)
2012 arXiv   pre-print
We present here algorithms for efficient computation 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  ... 
arXiv:1204.3735v1 fatcat:a3j26roivfd55fsf3sk7ahxgk4

Efficient matrix preconditioners for black box linear algebra

Li Chen, Wayne Eberly, Erich Kaltofen, B. David Saunders, William J. Turner, Gilles Villard
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  ... 
doi:10.1016/s0024-3795(01)00472-4 fatcat:4uc75rjqerborfo2bito2yn7jy

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

Jean-Guillaume Dumas , Pascal Giorgi
2007 arXiv   pre-print
This paper deals with 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.  ... 
arXiv:0704.2351v2 fatcat:rbmrycton5g7fkgee4lfxjb7ve

Parallel computation of the rank of large sparse matrices from algebraic K-theory

Jean-Guillaume Dumas, Philippe Elbaz-Vincent, Pascal Giorgi, Anna Urbánska
2007 Proceedings of the 2007 international workshop on Parallel symbolic computation - PASCO '07  
This paper deals with 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.  ... 
doi:10.1145/1278177.1278186 dblp:conf/issac/DumasEGU07 fatcat:tqci4jhlfnbv3eptehnioonq5i

Stein's method and the rank distribution of random matrices over finite fields

Jason Fulman, Larry Goldstein
2015 Annals of Probability  
With Q_q,n 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.  ... 
doi:10.1214/13-aop889 fatcat:7tnyo3dnjzedfhetqruklhisem

Multiple Target Localization in WSNs Based on Compressive Sensing Using Deterministic Sensing Matrices

Thu L. N. Nguyen, Yoan Shin
2015 International Journal of Distributed Sensor Networks  
Further simulation shows that 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.  ... 
doi:10.1155/2015/947016 fatcat:5xaxifvpnbc5dony6k4knpuzim
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