A 2-player game for adaptive covering codes

1. A 2-player game for adaptive covering codes Robert B. Ellis Texas A&M coauthors: Vadim Ponomarenko, Trinity University Catherine Yan, Texas A&M

2. A football pool problem Payoff: a bet with · 1 bad prediction Question. Min # bets to guarantee a payoff? Ans.=7

3. 11111 11110 11101 11011 10111 11111 10111 11100 11010 11001 10110 10101 10011 01111 01111 11000 11000 00100 10100 10010 10001 01110 01101 01011 00111 00010 00001 10000 01100 01010 01001 00110 00101 00011 01000 00100 00010 00001 00000 Covering code formulation W!1, L!0 C= Equivalent question What is the minimum number of radius 1 Hamming balls needed to cover the hypercube Q5?

4. Sparse history of covering code density

5. An adaptive football pool problem Payoff: a bet with · 1 bad prediction Question. Min # bets to guarantee a payoff? Ans.=6

6. Bets $ adaptive Hamming balls A “radius 1 bet” with predictions on 5 rounds can pay off in 6 ways: Round 5 Round 2 Round 4 Round 3 Round 1 A fixed choice in {0,1} for each “*” yields an adaptive Hamming ball of radius 1.

7. 11111 11110 11101 11011 10111 01111 11100 11010 11001 10110 10101 10011 01110 01101 01011 00111 11000 10100 10010 10001 01100 01010 01001 00110 00101 00011 10000 01000 00100 00010 00001 00000 Strategy tree for adaptive betting W/1 L/0 L/0 L/0 W/1 W/1 Paths to leaves containing 1: 11111 Root (0 incorrect predictions) 00101 Child 1 (1 incorrect prediction) 10101 Child 2  11001 Child 3  11101 Child 4  11110 Child 5 (1 incorrect prediction)

8. Adaptive covering code reformulation Definition. An adaptive (q,k)-code is a set of adaptive Hamming balls of radius k which cover the hypercube Qq. Theorem (E-P-Y). There exists a winning betting strategy for the q-round game with · k payoff-threshold iff there exists an adaptive (q,k)-code. Definition.Fk*(q) = minimum size adaptive (q,k)-code = minimum #bets for a winning betting strategy in q-rounds with · k payoff-threshold

9. The (x,q,k)*-game reformulation Players: Paul and Carole Parameters: q (#rounds), k, (x0,x1,…,xk), a nonneg. int. vec. Initial state:x=(x0,x1,…,xk) Game play: At an intermediate state x=(x0,x1,…,xk), a round consists of: a vector a=(a0,a1,…,ak), where 0 · ai· xi, chosen by Paul, and next state W(x,a)=(a0, a1+x0-a0, …, ak+xk-1-ak-1) or L(x,a)=(x0-a0, x1-a1+a0, …, xk-ak+ak-1) chosen by Carole. Determination of winner: After q rounds, Paul wins if the state vector is nonzero. Otherwise, Carole wins.

10. The Berlekamp weight function Restated Theorem (E-P-Y). Paul can win the ((x0,x1,…,xk),q,k)*-game iff there is a covering of Qq with xi adaptive Hamming balls of radius (k-i). Corollary. Fk*(q) = min size of an adaptive (q,k)-code = min n such that Paul can always win the ((n,0,…,0),q,k)-game. Definition (Berlekamp weight function). Intuition: when q rounds remain, the size of an adaptive Hamming ball of radius k is .

11. Conservation of weight lemma

12. Lower bound by probabilistic strategy

13. Upper bound: A counterexample W L 10 6 9 7 7 9 3-weight of possible next states

14. Upper bound: Perfect balancing 16 (4-weight) 8 (3-weight) 4 2 1

15. Upper bound: A balancing theorem

16. Upper bound: Main theorem

17. Upper bound: Stage I, x!y’

18. Upper bound: Stages I (con’t) & II

19. Upper bound: Stage III and conclusion

20. Exact result for k=1

21. Exact result for k=2

22. Linear relaxation and a random walk If Paul is allowed to choose entries of a to be real rather than integer, then a=x/2 makes the weight imbalance 0. Example: ((n,0,0,0),q,3)*-game and random walk on the integers:

23. Future directions • Efficient Algorithmic implementations of encoding/decoding using adaptive covering codes • Generalizations of the game to k a function of n • Generalization to an arbitrary communication channel(Carole has t possible responses, and certain responses eliminate Paul’s vector entirely) • Pullback of a directed random walk on the integers with weighted transition probabilities • Generalization of the game to a general weighted, directed graph • Comparison of game to similar processes such as chip-firing and the Propp machine via discrepancy analysis rellis@math.tamu.eduhttp://www.math.tamu.edu/~rellis/ vadim@trinity.eduhttp://www.trinity.edu/~vadim/ cyan@math.tamu.eduhttp://www.math.tamu.edu/~cyan/