The rank of random matrices over finite fields

A novel lower bound is introduced for the full rank probability of random finite field matrices, where a number of elements with known location are identically zero, and remaining elements are chosen independently of each other, uniformly over the field. The main ingredient is a result showing that constraining additional elements to be zero cannot result in a higher probability of full rank. The bound then follows by "zeroing" elements to produce a block-diagonal matrix, whose full rank probability can be computed exactly. The bound is shown to be at least as tight and can be strictly tighter than existing bounds.

[1]  Tor Bu.  Partitions of a vector space , 1980, Discret. Math..

[2]  M. Stephanov,et al.  Random Matrices , 2005, hep-ph/0509286.

[3]  Bartolomeu F. Uchôa Filho,et al.  On the Capacity of Multiplicative Finite-Field Matrix Channels , 2011, IEEE Transactions on Information Theory.

[4]  Tracey Ho,et al.  A Random Linear Network Coding Approach to Multicast , 2006, IEEE Transactions on Information Theory.

[5]  Colin Cooper,et al.  The cores of random hypergraphs with a given degree sequence , 2004, Random Struct. Algorithms.

[6]  A. A. Levitskaya.  Systems of Random Equations over Finite Algebraic Structures , 2005 .

[7]  Allan Sly,et al.  Random graphs with a given degree sequence , 2010, 1005.1136.

[8]  David Gamarnik,et al.  Combinatorial approach to the interpolation method and scaling limits in sparse random graphs , 2010, STOC '10.

[9]  Michael Molloy,et al.  The solution space geometry of random linear equations , 2011, Random Struct. Algorithms.

[10]  Jean Bourgain,et al.  On the singularity probability of discrete random matrices , 2009, 0905.0461.

[11]  Nicolas Macris,et al.  Spatial Coupling as a Proof Technique and Three Applications , 2013, IEEE Transactions on Information Theory.

[12]  Svante Janson,et al.  A simple solution to the k-core problem , 2007, Random Struct. Algorithms.

[13]  Michele Leone,et al.  Replica Bounds for Optimization Problems and Diluted Spin Systems , 2002 .

[14]  Joel H. Spencer,et al.  Sudden Emergence of a Giantk-Core in a Random Graph , 1996, J. Comb. Theory, Ser. B.

[15]  Gérard D. Cohen,et al.  The rate of regular LDPC codes , 2003, IEEE Trans. Inf. Theory.

[16]  Nicolas Macris,et al.  Threshold Saturation for Spatially Coupled LDPC and LDGM Codes on BMS Channels , 2013, IEEE Transactions on Information Theory.

[17]  F. Guerra.  Broken Replica Symmetry Bounds in the Mean Field Spin Glass Model , 2002, cond-mat/0205123.

[18]  Rüdiger L. Urbanke,et al.  Modern Coding Theory , 2008 .

[19]  D. Saad,et al.  Statistical mechanics of error-correcting codes , 1999 .

[20]  Colin Cooper,et al.  On the distribution of rank of a random matrix over a finite field , 2000, Random Struct. Algorithms.

[21]  Amin Coja-Oghlan,et al.  Harnessing the Bethe free energy† , 2015, Random Struct. Algorithms.

[22]  D. Panchenko.  SPIN GLASS MODELS FROM THE POINT OF VIEW OF SPIN DISTRIBUTIONS , 2010, 1005.2720.

[23]  E. Szemerédi,et al.  On the probability that a random ±1-matrix is singular , 1995 .

[24]  Boris G. Pittel,et al.  The Satisfiability Threshold for k-XORSAT , 2012, Combinatorics, Probability and Computing.

[25]  D Saad,et al.  Typical kernel size and number of sparse random matrices over Galois fields: a statistical physics approach. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[26]  Amir Dembo,et al.  Empirical Spectral Distributions of Sparse Random Graphs , 2016, Progress in Probability.

[27]  Yashodhan Kanoria,et al.  The set of solutions of random XORSAT formulae , 2011, SODA.

[28]  Marc Lelarge,et al.  The rank of diluted random graphs , 2010, SODA '10.

[29]  A. B. BASSET,et al.  Modern Algebra , 1905, Nature.

[30]  Robert G. Gallager,et al.  Low-density parity-check codes , 1962, IRE Trans. Inf. Theory.

[31]  John K. Reid,et al.  An Implementation of Tarjan's Algorithm for the Block Triangularization of a Matrix , 1978, TOMS.

[32]  M. Mézard,et al.  Two Solutions to Diluted p-Spin Models and XORSAT Problems , 2003 .

[33]  Oliver Riordan,et al.  The k-Core and Branching Processes , 2005, Combinatorics, Probability and Computing.

[34]  Michael Molloy,et al.  Cores in random hypergraphs and Boolean formulas , 2005, Random Struct. Algorithms.

[35]  Daniel Fernholz.  Cores and Connectivity in Sparse Random Graphs , 2004 .

[36]  Marc Lelarge,et al.  Bypassing correlation decay for matchings with an application to XORSAT , 2013, 2013 IEEE Information Theory Workshop (ITW).

[37]  Andrea Montanari,et al.  Tight Thresholds for Cuckoo Hashing via XORSAT , 2009, ICALP.

[38]  V. F. KOLCHIN.  Consistency of a system of random congruences , 1993 .

[39]  Jacob T. Schwartz,et al.  Fast Probabilistic Algorithms for Verification of Polynomial Identities , 1980, J. ACM.

[40]  Tracey Ho,et al.  Universal and robust distributed network codes , 2010, 2011 Proceedings IEEE INFOCOM.

[41]  Shenghao Yang,et al.  Symmetric Properties and Subspace Degradations of Linear Operator Channels over Finite Fields , 2011, ArXiv.

[42]  V. F. Kolchin,et al.  Random Graphs and Systems of Linear Equations in Finite Fields , 1994, Random Struct. Algorithms.

[43]  Andreas Goerdt,et al.  Satisfiability Thresholds beyond k -XORSAT , 2012, CSR.

[44]  Dana Moshkovitz.  An Alternative Proof of The Schwartz-Zippel Lemma , 2010, Electron. Colloquium Comput. Complex..

[45]  Vijaya Ramachandran,et al.  The giant k-core of a random graph with a speci ed degree sequence , 2003 .

[46]  Andrea Montanari,et al.  Tight bounds for LDPC and LDGM codes under MAP decoding , 2004, IEEE Transactions on Information Theory.

[47]  D. Burshtein,et al.  Upper bounds on the rate of LDPC codes , 2002, Proceedings IEEE International Symposium on Information Theory,.

[48]  Vasek Chvátal,et al.  Almost All Graphs with 1.44n Edges are 3-Colorable , 1991, Random Struct. Algorithms.

[49]  Ian F. Blake,et al.  Windowed Erasure Codes , 2006, 2006 IEEE International Symposium on Information Theory.

[50]  R. Monasson,et al.  Rigorous decimation-based construction of ground pure states for spin-glass models on random lattices. , 2002, Physical review letters.

[51]  V. F. KOLCHIN,et al.  On the number of cycles in a random non-equiprobable graph , 1992 .

[52]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[53]  Jeong Han Kim,et al.  Poisson Cloning Model for Random Graphs , 2008, 0805.4133.

[54]  Igor N. Kovalenko,et al.  On the Limit Distribution of the Number of Solutions of a Random System of Linear Equations in the Class of Boolean Functions , 1967 .

[55]  T. Tao,et al.  On the singularity probability of random Bernoulli matrices , 2005, math/0501313.

[56]  Colin Cooper,et al.  On the rank of random matrices , 2000, Random Struct. Algorithms.

[57]  Alex J. Grant,et al.  Quasi-Uniform Codes and Their Applications , 2013, IEEE Transactions on Information Theory.

[58]  Richard M. Karp,et al.  The rank of sparse random matrices over finite fields , 1997, Random Struct. Algorithms.

[59]  Tomasz Luczak,et al.  Size and connectivity of the k-core of a random graph , 1991, Discret. Math..

[60]  S. Kak.  Information, physics, and computation , 1996 .