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

A football pool problem Round 1Round 2Round 3Round 4Round 5 Bet 1WWWWW Bet 2LWWWW Bet 3WLWWW Bet 4WWLLL Bet 5LLWLL Bet 6LLLWL Bet 7LLLLW Payoff: a bet with · 1 bad prediction Question. Min # bets to guarantee a payoff? Ans.=7

Covering code formulation W ! 1, L ! 0 Equivalent question What is the minimum number of radius 1 Hamming balls needed to cover the hypercube Q 5 ? C=C=

Sparse history of covering code density

An adaptive football pool problem Round 1Round 2Round 3Round 4Round 5 Bet 1W Bet 2W Bet 3W Bet 4L Bet 5L Bet 6L Actual Payoff: a bet with · 1 bad prediction Question. Min # bets to guarantee a payoff? Ans.=6

Round 1 Bets $ adaptive Hamming balls A “radius 1 bet” with predictions on 5 rounds can pay off in 6 ways: Root11010All predictions correct Child 10****1 st prediction incorrect Child 210***2 nd prediction incorrect Child 3111**3 rd prediction incorrect Child 41100*4 th prediction incorrect Child th prediction incorrect Round 2 Round 3 Round 4 Round 5 A fixed choice in {0,1} for each “*” yields an adaptive Hamming ball of radius 1.

Strategy tree for adaptive betting W/1L/0 W/1 L/0 W/1 L/0 Paths to leaves containing 1: 11111Root (0 incorrect predictions) 00101Child 1 (1 incorrect prediction) 10101Child 2  11001Child 3  11101Child 4  11110Child 5 (1 incorrect prediction)

Adaptive covering code reformulation Definition. An adaptive (q,k)-code is a set of adaptive Hamming balls of radius k which cover the hypercube Q q. 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. F k * (q) = minimum size adaptive (q,k)-code = minimum #bets for a winning betting strategy in q-rounds with · k payoff-threshold

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

The Berlekamp weight function Restated Theorem (E-P-Y). Paul can win the ((x 0,x 1,…,x k ),q,k) * -game iff there is a covering of Q q with x i adaptive Hamming balls of radius (k-i). Corollary. F k * (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.

Conservation of weight lemma

Lower bound by probabilistic strategy

Upper bound: A counterexample weight of possible next states WL

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

Upper bound: A balancing theorem

Upper bound: Main theorem

Upper bound: Stage I, x ! y’

Upper bound: Stages I (con’t) & II

Upper bound: Stage III and conclusion

Exact result for k=1

Exact result for k=2

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:

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