A 2-player game for adaptive covering codes

1
A 2-player game for adaptive covering codes

• Robert B. Ellis
• Texas AM
• coauthors
• Vadim Ponomarenko, Trinity University
• Catherine Yan, Texas AM

2
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
prediction Question. Min bets to guarantee a
payoff?
Ans.7
3
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
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
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

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
Round 2
Round 4
Round 5
Round 3
Round 1
A fixed choice in 0,1 for each yields an
adaptive Hamming ball of radius 1.
7
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, a1x0-a0, , akxk-1-ak-1) or
L(x,a)(x0-a0, x1-a1a0, , xk-akak-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 (cont) II
19
Upper bound Stage III and conclusion
20
Exact result for k1
21
Exact result for k2
22
Linear relaxation and a random walk
If Paul is allowed to choose entries of a to be
real rather than integer, then ax/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 Pauls 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_at_math.tamu.edu http//www.math.tamu.edu/rel
lis/ vadim_at_trinity.edu http//www.trinity.edu/va
dim/ cyan_at_math.tamu.edu http//www.math.tamu.edu/
cyan/