Coding Theory: Packing, Covering, and 2-Player Games Robert Ellis Menger Day 2008: Recent Applied Mathematics Research Advances April 14, 2008

2 Overview  A problem from integrated circuit design  Coding Theory –Error-correcting codes and packings –Error-correcting codes as a 2-player liar game –Covering codes –Covering codes as a football pool  Coding with Feedback –A liar game and an adaptive football pool –Near-perfect radius 1 adaptive codes  Results and Research Questions in Liar Games

3 A VLSI Layout Problem Silicon substrate Wires & components Inert metal fill Fill Library 2 6 patterns  2 3 patterns, Compression ratio: 50%

4 An Asymmetric Covering Code  Fill library  (6,2)-asymmetric binary code  Size bound  (2 n /n R ) (Cooper,Ellis,Kahng `02)  Application to VLSI Layout (Ellis,Kahng,Zheng `03)  Improved fixed-parameter codes: Applegate,Rains,Sloane `03; Exoo `04; Östergård,Seuranen `04  Improved size bound (Krivelevich,Sudakov,Vu `03) Codeword:

5 Therefore K + (4,2) = 6 (length=4, radius=2). Smallest (4,1)-Asymmetric Covering Code

6 00…00 11…11  Select each word to be in the code with probability p(n)  Any uncovered word is added as a codeword  This plus hypercube structure yields codes of size  (2 n /n R )  Best possible up to a constant, since middle ball volumes are  (n R ) Good (n,R)-Asymmetric Covering Codes

7 Coding Theory Overview  Coding theory concerns the properties of sets of codewords, or fixed-length strings from a finite alphabet.  Primary uses: Error-correction for transmission in the presence of noise Compression of data with or without loss  Many viewpoints afforded: Packings and coverings of Hamming balls in the n -cube 2-player perfect information games

8  Noisy communication: add redundancy to counteract noise  Noiseless communication: compress data using redundancy  The binary symmetric channel for noise 0 ≤ p < 1/2 Information Theory (Shannon Model) sender receiver encoder decoder Noise  1 …  n x1…xnx1…xn (x 1 +  1 )…(x n +  n ) m m Claude Shannon p p 1-p

9  Transmit blocks of length n  Noise changes ≤ e bits per block ( ||  || 1 ≤ e )  Repetition code 111, 000 – length: n = 3 – e = 1 –information rate: 1/3 Coding Theory: (n,e) -Codes  x1…xnx1…xn (x 1 +  1 )…(x n +  n ) Received: Decoded: blockwise majority vote Richard Hamming

10 Block Codes from now on Restricting to block codes still includes  Convolutional codes (cell phones, Bluetooth)  Reed-Solomon codes (CDs, DSL, WiMAX)  Turbo codes (Mars Reconnaissance Orbiter) (assumptions on noise for these codes will vary)

errors: incorrect decoding Coding Theory – A Hamming (7,1)-Code Length n=7, corrects e=1 error received decoded error: correct decoding

12  (3,1)-code: 111, 000  Pairwise distance = 3  1 error can be corrected  The M codewords of an (n,e) -code correspond to a packing of Hamming balls of radius e in the n -cube A Repetition Code as a Packing A packing of 2 radius 1 Hamming balls in the 3-cube

13 A (5,2) -Code as a Packing (5,2)-code: 01100, (disjoint) packing in 5-cube Volume: Sphere Bound: for an (n,e) - code with M codewords,

14  (5,1)-code: 11111, 10100, 01010, A (5,1) -Code as a 2-Player Game 0What is the 5 th bit? 1What is the 4 th bit? 0What is the 3 rd bit? 0What is the 2 nd bit? 0What is the 1 st bit? CarolePaul >1 # errors

15  Covering is the companion problem to packing  Packing: (n,e) -code  Covering: (n,R) -code Covering Codes length packing radius covering radius (3,1) -packing code and (3,1) -covering code “perfect code” (5,1)-packing code(5,1)-covering code

16 Optimal Length 5 Packing & Covering Codes (5,1) -packing code (5,1) -covering code

17 A (5,1) -Covering Code as a Football Pool WLLLLBet 7 LWLLLBet 6 LLWLLBet 5 LLLWWBet 4 WWWLWBet 3 WWWWLBet 2 WWWWWBet 1 Round 5Round 4Round 3Round 2Round 1 Payoff: a bet with ≤ 1 bad prediction Question. Min # bets to guarantee a payoff? Ans.=

18 Codes with Feedback (Adaptive Codes) sender receiver Noise Noiseless Feedback Elwyn Berlekamp  Feedback Noiseless, delay-less report of actual received bits  Improves the number of decodable messages E.g., from 20 to 28 messages for an (8,1) -code 1, 0, 1, 1, 0 1, 1, 1, 1, 0

19 A (5,1) -Adaptive Code as a 2-Player Liar Game A D B C 0 1 >1 # lies YIs the message C? NIs the message D? NIs the message B? NIs the message A or C? YIs the message C or D? CarolePaul Message Original encoding Adapted encoding A B C D **** 11*** 10*** 1000* 111**100** 1000* Y  1, N  0

20 A (5,1)-Adaptive Covering Code as a Football Pool LWLLW Carole L Bet 6 L Bet 5 L Bet 4 W Bet 3 W L L WW Bet 2 L W W W W W L L WW Bet 1 Round 5Round 4Round 3Round 2Round 1 Payoff: a bet with ≤ 1 bad prediction Question. Min # bets to guarantee a payoff? Ans.=6 Bet 3 Bet 6 Bet 4 Bet >1 # bad predictions (# lies) Bet 2 Bet 1

21  Form of an adaptive Hamming ball (radius 1)  Example: n = 5, e = R = 1 Feedback and Adaptive Hamming Balls Adapted encoding after * * * * * * 1 Error in 5 th bit 1 Child 5 Error in 4 th bit * Child 4 Error in 3 rd bit * Child 3 Error in 2 nd bit * Child 2 Error in 1 st bit * Child 1 Original encoding 0 Root

22 Classification of Coding Problems Packing No feedback Error-correcting codes P(n,e) Feedback Adaptive error- correcting codes P’(n,e) Covering Feedback Adaptive covering codes K’(n,R) No feedbackCovering codes K(n,R) Sphere Bound ≤ ≤ ≤ ≤

23 Near-Perfect Radius 1 Adaptive Codes Theorem (E.`05+). For all n ≥ 2 and e = R = 1, there exists an adaptive packing contained in an adaptive covering with sizes given by where (The sphere bound is ) P’(n,1)K’(n,1)

24 Proof Idea: Near-Perfect Radius 1 Adaptive Codes packing covering steal Q1Q1 Q2Q2 duplicate 0Q 1 1Q 1 duplicate Q2Q steal 0Q 2 1Q Q3Q Q3Q Q 2 packing Q 2 covering

25 Adaptive Coding as an (M,n,e) -Liar Game  M = # chips n = # rounds e = max # lies Carole picks a distinguished x 2 {1,…,M} 1 M >e>e # lies … e … (1) Paul bipartitions {1,…,M} = A 0 [ A 1 and asks “Is x 2 A 1 ?” (2) Carole responds “Yes” or “No”, and may lie up to e times. Each Round > A0A0 A1A1 “Yes” “No” >

26 Lose Original and Pathological Liar Games  Two variants –Original liar game (Berlekamp, Rényi, Ulam) Paul wins iff at most 1 chip survives after n rounds –Pathological liar game (Ellis&Yan) Paul wins iff at least 1 chip survives after n rounds … 0 1 >3 2 3 … … LoseWin … 0 1 >3 2 3 … … Win

27 Adaptive error- correcting codes Liar game Classification of Coding Problems Covering codes Adaptive covering codes Error-correcting codes K(n,R) No feedback K’(n,R) Feedback Covering P’(n,e) Feedback P(n,e) No feedback Packing Sphere Bound ≤ ≤ ≤ ≤ Pathological liar game

28 3 Chip Original Liar Game  Given M=3 chips, in how many rounds can Paul guarantee winning the game with e lies?  Label each chip with its distance to being eliminated  Introduce weight function f(x 1,x 2,x 3 )=x 1 +x 2 +x 3 -1 f(6,4,3) = 12  Each round: –Paul can force f to reduce by 1 –Carole can prevent f from reducing by more than >e>e e … > A0A0 A1A1 Paul wins iff n ≥ 3e+2

Chip Original Liar Game Order chip labels so that x 1 ≥ x 2 ≥ … ≥ x M.  M=4 chips: f 4 (x 1,x 2,x 3,x 4 )=x 1 +x 2 +x 3 -1 = 12  M=5 chips: f 5 (x 1,x 2,x 3,x 4,x 5 )=x 1 +x 2 +x 3 +  (x 1 =x 5 )-1  Exercise: find/verify the weight function for M=4,…,8 (Ellis&Łuczak)  Research Problem: find the weight function for M> > A0A0 A1A f 5 =18+1-1= > f 5 =18+0-1=17 (f 3 =18-1=17)

Chip Pathological Liar Game  M=2 chips g 2 (x 1,x 2 )=x 1 +x 2 -1 = 9  M=3 chips g 3 (x 1,x 2,x 3 )=x 1 +x 2 -1 = 9  M=4 chips g 4 (x 1,x 2,x 3,x 4 )=x 1 +x 2 +  (x 1 =x 4 )-1 = 12  M=2,3,4 (Ellis&Stanford) M>4: Research Problem > A0A0 A1A g 4 =12+1-1= > g 4 =12+0-1=11 (g 2 =12-1=11)

31 Perfect Splits and the Pathological Liar Game 0 1 >e>e e k-1 k 2k2k … … … 0 1 >e>e e k 2 k-1 … … … 0 1 >e>e e k-1 k 1 … … … 1 1 round k-1 rounds 0 1 >e>e e k-1 k 0 … … … Remove chips 2 k’ 0 1 >e>e e k-1 k 0 … … … Repeat until 1 chip left at position e … …

32  Upper bound on M=2 k : e/n is the overall fraction of lies  after k rounds the chips at position (e/n)k determine whether Paul wins  Lower bound on M=2 k : Each chip survives in only out of 2 n possible outcomes of the game; i.e., Perfect Splits and the Pathological Liar Game 0 1 >e>e e k-1 k 0 … … … k’ 0 error fraction

33 Many Open Questions at Every Level!  Research problems appropriate for Undergraduates, Graduate students, Dissertations, and beyond!  Fixed parameter games  Games with constrained lies  Non-binary alphabets  Restricted feedback  List decoding (win with L chips instead of 1)  Applying feedback coding to real-world problems