Corpus ID: 141481852

# How to Find New Characteristic-Dependent Linear Rank Inequalities using Binary Matrices as a Guide

@article{Pea2019HowTF,
title={How to Find New Characteristic-Dependent Linear Rank Inequalities using Binary Matrices as a Guide},
author={Victor Pe{\~n}a and Humberto Sarria},
journal={ArXiv},
year={2019},
volume={abs/1905.00003}
}
• Published 29 April 2019
• Computer Science, Mathematics
• ArXiv
In Linear Algebra over finite fields, a characteristic-dependent linear rank inequality is a linear inequality that holds by ranks of subspaces of a vector space over a finite field of determined characteristic, and does not in general hold over other characteristics. In this paper, we show a method to produce these inequalities using binary matrices with suitable ranks over different fields. In particular, for each $n\geq7$, we produce \$2\left\lfloor \frac{n-1}{2}\right\rfloor -4… Expand
1 Citations

#### Figures and Topics from this paper

Common information, matroid representation, and secret sharing for matroid ports
• Computer Science, Mathematics
• Des. Codes Cryptogr.
• 2021
Improved results have been obtained in recent works by using the properties from which they are derived instead of the inequalities themselves to the classification of matroids according to their representations and the search for bounds on secret sharing for matroid ports. Expand

#### References

SHOWING 1-10 OF 10 REFERENCES
Characteristic-dependent linear rank inequalities via complementary vector spaces
• Computer Science, Mathematics
• ArXiv
• 2019
This paper produces new characteristic- dependent linear rank inequalities by an alternative technique to the usual Dougherty's inverse function method, and shows that there exists a sequence of networks in which each member is linearly solvable over a field if and only if the characteristic of the field is in P, and the linear capacity over fields whose characteristic is not in P. Expand
Linear rank inequalities on five or more variables
• Mathematics, Computer Science
• ArXiv
• 2009
It is proved that there are essentially new inequalities at each number of variables beyond four (a result also proved recently by Kinser) and a list of 24 which, together with the Shannon and Ingleton inequalities, generate all linear rank inequalities on five variables is given. Expand
Characteristic-Dependent Linear Rank Inequalities With Applications to Network Coding
• Mathematics, Computer Science
• IEEE Transactions on Information Theory
• 2015
Two characteristic-dependent linear rank inequalities are given for eight variables and applications of these inequalities to the computation of capacity upper bounds in network coding are demonstrated. Expand
New inequalities for subspace arrangements
• R. Kinser
• Mathematics, Computer Science
• J. Comb. Theory, Ser. A
• 2011
For each positive integer n>=4, this work gives an inequality satisfied by rank functions of arrangements of arrangement of n subspaces that can be thought of as a hierarchy of necessary conditions for a (poly)matroid to be realizable. Expand
Lexicographic Products and the Power of Non-linear Network Coding
• Computer Science, Mathematics
• 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science
• 2011
The technique uses linear programs to establish separations between combinatorial and coding-theoretic parameters and applies hyper graph lexicographic products to amplify these separations, showing a polynomial separation between linear and non-linear network coding rates. Expand
Inequalities for Shannon Entropy and Kolmogorov Complexity
• Mathematics, Computer Science
• J. Comput. Syst. Sci.
• 2000
An inequality for Kolmogorov complexities that implies Ingleton's inequality for ranks is presented and another application is a new simple proof of one of Gacs-Korner's results on common information. Expand
Achievable Rate Regions for Network Coding
• Computer Science, Mathematics
• IEEE Transactions on Information Theory
• 2012
In addition to the known matrix-computation method for proving outer bounds for linear coding, a new method is presented that yields actual characteristic-dependent linear rank inequalities from which the desired bounds follow immediately. Expand
Representation of matroids
• Mathematics
• 2002
In this paper we give a necessary and sufficient criterion for representability of a matroid over an algebraic closed field. This leads to an algorithm, based on an extension of Groebner Bases, inExpand
Network routing capacity
• Mathematics, Computer Science
• IEEE Transactions on Information Theory
• 2005
The routing capacity of a network is defined to be the supremum of all possible fractional message throughputs achievable by routing, and it is proved that every rational number in (0, 1] is the routingcapacity of some solvable network. Expand
Insufficiency of linear coding in network information flow
• Mathematics, Computer Science
• Proceedings. International Symposium on Information Theory, 2005. ISIT 2005.
• 2005
It is shown that the network coding capacity of this counterexample network is strictly greater than the maximum linear coding capacity over any finite field, so the network is not even asymptotically linearly solvable. Expand