A 2-player game for adaptive covering codes

Robert B. EllisTexas A&M

coauthors:Vadim Ponomarenko, Trinity University

Catherine Yan, Texas A&M

A football pool problem

Round 1 Round 2 Round 3 Round 4 Round 5

Bet 1 W W W W W

Bet 2 L W W W W

Bet 3 W L W W W

Bet 4 W W L L L

Bet 5 L L W L L

Bet 6 L L L W L

Bet 7 L L L L W

Payoff: a bet with · 1 bad predictionQuestion. Min # bets to guarantee a payoff?

Ans.=7

Covering code formulation11111

11101 11011 1011111110

11100 11010 11001 10110 10101 10011

10100 10010 1000111000

10000

01111

01101 01011 0011101110

01100 01010 01001 00110 00101 00011

00100 00010 0000101000

00000

W!1, L!0

Equivalent questionWhat is the minimum number of radius 1 Hamming balls needed to cover the hypercube Q5?

1111110111

1100001111

001000001000001

C=

Sparse history of covering code density

An adaptive football pool problem

Round 1 Round 2 Round 3 Round 4 Round 5

Bet 1 W

Bet 2 W

Bet 3 W

Bet 4 L

Bet 5 L

Bet 6 L

Actual

Payoff: a bet with · 1 bad predictionQuestion. Min # bets to guarantee a payoff?

Ans.=6

Rou

nd 1

Bets $ adaptive Hamming ballsA “radius 1 bet” with predictions on 5 rounds can pay off in 6 ways:

Root 1 1 0 1 0 All predictions correct

Child 1 0 * * * * 1st prediction incorrect

Child 2 1 0 * * * 2nd prediction incorrect

Child 3 1 1 1 * * 3rd prediction incorrect

Child 4 1 1 0 0 * 4th prediction incorrect

Child 5 1 1 0 1 1 5th prediction incorrect

Rou

nd 2

Rou

nd 3

Rou

nd 4

Rou

nd 5

A fixed choice in {0,1} for each “*” yields an adaptive Hamming ball of radius 1.

Strategy tree for adaptive betting

W/1 L/0

W/1 L/0 W/1 L/0

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)

11011 10111

11100 11010 10110 10011

10100 10010 1000111000

10000

01111

01101 01011 0011101110

01100 01010 01001 00110

11111

1110111110

11001 10101

00101 00011

00100 00010 0000101000

00000

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

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.

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 .

Conservation of weight lemma

Lower bound by probabilistic strategy

Upper bound: A counterexample

10 6 9 7 7 9

3-weight of possible next states

W L

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 walkIf 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

rellis@math.tamu.edu http://www.math.tamu.edu/~rellis/

vadim@trinity.edu http://www.trinity.edu/~vadim/

cyan@math.tamu.edu http://www.math.tamu.edu/~cyan/