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 BACn =

• Pn

p,q | Pp,q is a BAC

• ,

be the set of all n dimensional BACs. For a fixed n ∈ N :

• Given Pn

p,q ∈ BACn we construct a bi-oriented weighted graph

G = (Fn

2, E);

• From G, the usual path-lenght determines a quasi-metric

δ(X, Y ) that is matched to the corresponding Pn

p,q.

Definition Given two channels Pn

p,q and Pn p′,q′, Pn p,q ∼ Pn p′,q′ if the

inequality holds Pn

p,q(X|Y ) < Pn p,q(X|Z) ⇐

⇒ Pn

p′,q′(X|Y ) < Pn p′,q′(X|Z),

for all X, Y , Z ∈ Fn

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

2

• 1 + n

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), no 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.