Repairing Four-Atom Conjecture Ting-Ting Nan Advisor: Nigel Boston - PowerPoint PPT Presentation

Repairing Four-Atom Conjecture Ting-Ting Nan Advisor: Nigel Boston SP Coding and Information School Ting-Ting Nan Repairing Four-Atom Conjecture

Figure: Butterfly network. Ting-Ting Nan Repairing Four-Atom Conjecture

Entropy Region Entropy is a measure of the information/uncertainty in a random variable. Shannon entropy H ( X ) for a discrete random variable X over alphabet A is � − p ( x ) log p ( x ) = − E X [log p ( X )] . x ∈A The region Γ ∗ n is defined to consist of all entropic vectors of any n discrete random variables. Ting-Ting Nan Repairing Four-Atom Conjecture

Ingleton Inequality I ( X 1 ; X 2 | X 3 ) + I ( X 1 ; X 2 | X 4 ) − I ( X 1 ; X 2 ) ≥ 0 . Facts: The Ingleton inequality does NOT hold always. All linear codings satisfy the Ingleton inequality. All vectors inside Γ 4 which satisfy the Ingleton inequality are entropic. The Ingleton score is I ( X 1 ; X 2 ) − I ( X 1 ; X 2 | X 3 ) − I ( X 1 ; X 2 | X 4 ) . H 1234 Ting-Ting Nan Repairing Four-Atom Conjecture

The Four-Atom Conjecture Conjecture (The Four-Atom Conjecture) The Ingleton score s can not exceed 0.089373. a a R. Dougherty, C. Freiling, and K. Zeger, Insufficiency of linear network coding in network information flow, in IEEE Transactions on Information Theory, 2005. Ting-Ting Nan Repairing Four-Atom Conjecture