Causal adversarial channel codes OPEN UNIVERSITY OF ISRAEL

Causal adversarial channel codes OPEN UNIVERSITY OF ISRAEL Bikash Dey Sidharth Jaggi Michael Langberg

Noisy Channels 1 1 q=2 (binary) “large” q “Folklore” [Sha 48] R R p um xn 0. 5 p 1 yn en pn ûm =um Alphabet q R=m/n

Adversarial Channels Noisy 1 R 1 q=2 (binary) “large” q [Reed-Solomon] [Mc. ERRW 77] R [Gilbert Varshamov] 0. 25 p 0. 5 1 Adversarially chosen

Causal (online) Adversarial Channels Causally adversarially chosen

ISIT 2009 [DJL] q=2 (binary) 0 -delay Intriguingly intricate… Random Plotkin Bayes’ Turan’s Coding bound Theorem theorem

Large-q results 1 1. 0 -delay R p 0. 5 1

Large-q results d 1 1. 0 -delay 2. d-delay additive R p 1

Large-q results d 1 1. 0 -delay 2. d-delay additive 3. d-delay overwrite d R p Tight rate-regions, Poly-time algorithms. 0. 5 1

1. 0 -delay Wait-and-push If R=(1 -2 p)++ε n(1 -2 p)+ np np ?

1. 0 -delay If R=(1 -2 p)+, Reed-Solomon codes

2. d-delay additive ? pn-erasure correcting code

3. d-delay overwrite 0 -delay: Wait-and-push n(1 -2 p)+ np np d-delay: Wait-babble-and-push n(1 -2 p+d)+ nd n(p-d)

3. d-delay overwrite DOODAD: (noun) A small specialized mechanical device: concern, contraption, Delay Online Overwriting Disruptive Adversary Decoding (Algorithm) contrivance, gadget, gimmick, jigger, thing. Informal doohickey, widget. Slang gizmo. See machine.

3. d-delay overwrite DOODAD: nd nd

3. d-delay overwrite DOODAD:

3. d-delay overwrite DOODAD: 1. p

3. d-delay overwrite DOODAD: 2. 0. 5>p>d Longest connected chain nd nd A. # correct chains B. # symbols in correct chains C. Redundancy in erasure-correcting code Poly-time rate-optimal algorithm

Summary/thoughts n n Large alphabets (now) understood, small not… Causal adversary still pretty strong… … but delays can weaken him. Types of error matter (additive/overwrite/…? )

Questions?