Characterization of the Space of Binary Asymmetric Channels (BACs) - PowerPoint PPT Presentation

Characterization of the Space of Binary Asymmetric Channels (BACs) Christiane Buffo Rodrigues chrismmor@gmail.com Advisor: Marcelo Firer mfirer@ime.unicamp.br IMECC - Unicamp

Introduction Let BAC n = P n � � p , q | P p , q is a BAC , be the set of all n dimensional BACs. For a fixed n ∈ N : p , q ∈ BAC n we construct a bi-oriented weighted graph • Given P n G = ( F n 2 , E ); • From G , the usual path-lenght determines a quasi-metric δ ( X , Y ) that is matched to the corresponding P n p , q . Definition Given two channels P n p , q and P n p ′ , q ′ , P n p , q ∼ P n p ′ , q ′ if the inequality holds P n p , q ( X | Y ) < P n ⇒ P n p ′ , q ′ ( X | Y ) < P n p , q ( X | Z ) ⇐ p ′ , q ′ ( X | Z ) , for all X , Y , Z ∈ F n 2 .

Results Fixing n ∈ N , • The Equivalence Classes of BACs are described considering the homeomorphism � � Φ( p , q ) = log(1 / p − 1) , log(1 / q − 1) ; • Since n is fixed, we get a partition of ( R + × R + ) into � � 1 + � n 2 i =2 φ ( i ) convex cones; • Each cone corresponds to a unique Equivalence Class of BACs; • According to the NN decoding criteria we may determine the number of different (quasi-)metric balls centered at X ; • The description of the balls depends on the cone, because each cone determines a decoding.

References J.L. Massey, Notes on coding theory , class notes for course 6.575 (spring), M.I.T., Cambridge, MAA, 1967. G. S´ eguin, On metrics matched to the discrete memoryless channel , J. Franklin Inst. 309 (1980), n o 3, 179-189. M.M. Deza, E. Deza, Encyclopedia of distances , second ed., Springer, Heidelberg, 2013. A. Fazeli, A. Vardy, E. Yaakobi, Generalized Sphere Packing bound: Applications , 2014 IEEE International Symposium on Information Theory (ISIT), 1261-1265.