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 −

• x∈A

p(x) log p(x) = −EX[log p(X)]. 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(X1; X2|X3) + I(X1; X2|X4) − I(X1; X2) ≥ 0. Facts: The Ingleton inequality does NOT hold always. All linear codings satisfy the Ingleton inequality. All vectors inside Γ4 which satisfy the Ingleton inequality are entropic. The Ingleton score is I(X1; X2) − I(X1; X2|X3) − I(X1; X2|X4) H1234 .

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

• aR. 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