The satisfiability threshold for random linear equations
| Autor: | Noela Müller, Amin Coja-Oghlan, Peter J. Ayre, Pu Gao |
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| Jazyk: | angličtina |
| Rok vydání: | 2017 |
| Předmět: |
FOS: Computer and information sciences
Discrete Mathematics (cs.DM) Multivariate random variable Linear system 05C80 Combinatorial proof Second moment of area Satisfiability Combinatorics Computational Mathematics Matrix (mathematics) Finite field FOS: Mathematics Discrete Mathematics and Combinatorics Mathematics - Combinatorics Combinatorics (math.CO) Linear equation Mathematics Computer Science - Discrete Mathematics |
| Popis: | Let $A$ be a random $m\times n$ matrix over the finite field $F_q$ with precisely $k$ non-zero entries per row and let $y\in F_q^m$ be a random vector chosen independently of $A$. We identify the threshold $m/n$ up to which the linear system $A x=y$ has a solution with high probability and analyse the geometry of the set of solutions. In the special case $q=2$, known as the random $k$-XORSAT problem, the threshold was determined by [Dubois and Mandler 2002, Dietzfelbinger et al. 2010, Pittel and Sorkin 2016], and the proof technique was subsequently extended to the cases $q=3,4$ [Falke and Goerdt 2012]. But the argument depends on technically demanding second moment calculations that do not generalise to $q>3$. Here we approach the problem from the viewpoint of a decoding task, which leads to a transparent combinatorial proof. |
| Databáze: | OpenAIRE |
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