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

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

4. 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) 000000 000010010100 111000 000001 000011101100 010111 011101 Codeword: 010100

5. 5 Therefore K + (4,2) = 6 (length=4, radius=2). Smallest (4,1)-Asymmetric Covering Code 010111000011011010011010 1011011111011110 1111 0000 0100001000011000 0000 0100001000011000110010011010 1000 010111000110 01000011 10110111 1100 11011110 1111

6. 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. 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. 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 0 0 1 1 p p 1-p

9. 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 ) 110010000 101 000111 Received: Decoded: blockwise majority vote Richard Hamming

10. 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)

11. 11 0010011 3 errors: incorrect decoding Coding Theory – A Hamming (7,1)-Code 10001110110110 01000110101101 00101010011011 00011101110001 00000001101010 11001001011100 10100100111000 10010011111111 Length n=7, corrects e=1 error 1001011 received decoded 1001001 1 error: correct decoding

12. 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 110011101 111 000 010001100 000 010001100 110011101 111 A packing of 2 radius 1 Hamming balls in the 3-cube

13. 13 A (5,2) -Code as a Packing 0100101100 0111001101 00100 11100 01000 111101110101111 00000 0101011000101000011000101 10110 10011 1000110010110110001110111 00001000101000011111101010011101011110011100111010 (5,2)-code: 01100, 10011 (disjoint) packing in 5-cube Volume: Sphere Bound: for an (n,e) - code with M codewords,

14. 14  (5,1)-code: 11111, 10100, 01010, 00001 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 11111 00001 10100 01010 0 1 >1 # errors 11111000011010001010 01111001000001000011 00100 01010 00010 00001 11111 101000101000001

15. 15  Covering is the companion problem to packing  Packing: (n,e) -code  Covering: (n,R) -code Covering Codes length packing radius covering radius 110011101 111 000 010001100 000 010001100 110011101 111 (3,1) -packing code and (3,1) -covering code “perfect code” 11111 00001 10100 01010 11111 11000 01111 1011100001 00100 00010 (5,1)-packing code(5,1)-covering code

16. 16 Optimal Length 5 Packing & Covering Codes 01001 01100 01110 01101 00100 11100 01000 111101110101111 00000 01010 11000 10100 00110 00101 10110 10011 10001 10010 11011 00011 10111 000010001010000 11111 10101 00111 01011 11001 11010 0111001101 0100101100 00100 11100 01000 111101110101111 00000 0101011000101000011000101 1011010011 1000110010 11011 00011 10111 000010001010000 11111 1010100111010111100111010 (5,1) -packing code (5,1) -covering code

17. 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.=7 00100 01111 11000 10111 00001 00010 11111

18. 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. 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 00101 Message Original encoding Adapted encoding A B C D 0111001010 11000 10011 1**** 11*** 10*** 1000* 111**100** 1000* 1000010001 Y  1, N  0

20. 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 5 0 1 >1 # bad predictions (# lies) Bet 2 Bet 1

21. 21  Form of an adaptive Hamming ball (radius 1)  Example: n = 5, e = R = 1 Feedback and Adaptive Hamming Balls Adapted encoding after 1 1 1 1 0 1 1 1 1 0 * 1 0 0 1 * * 0 1 0 * * * 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 01010 11010 1000101111111001100111011

22. 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. 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. 24 Proof Idea: Near-Perfect Radius 1 Adaptive Codes 0101 1 00 10 packing covering 0 1 00 0101 10 1 steal 00 0101 10 1 Q1Q1 Q2Q2 duplicate 0Q 1 1Q 1 duplicate 00 0101 10 1 Q2Q2 000 001 010 011011 100 101 110 111111 steal 0Q 2 1Q 2 000 001 010 011011 100 101 110 111111 Q3Q3 000 001 010 011011 100 101 110 111111 Q3Q3 0101 1 00 10 Q 2 packing Q 2 covering

25. 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 2 0 1 >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 9 0 1 >3 2 3 3 2 1 4 6 5 8 7 A0A0 A1A1 “Yes” “No” 9 0 1 >3 2 3 4 2 1 3 6 5 8 7 9 0 1 2 3 3 2 1 4 6 5 8 7

26. 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 … 0 1 2 3 … 0 1 2 3 LoseWin … 0 1 >3 2 3 … 0 1 2 3 … 0 1 2 3 Win

27. 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. 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 1 0 1 >e>e e … 6 0 1 >5 5 3 4 2 34 A0A0 A1A1 Paul wins iff n ≥ 3e+2

29. 29 4-8 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>8 6 0 1 >5 5 3 4 2 34 A0A0 A1A1 1 6 0 1 5 2 34 6 6 6 6 f 5 =18+1-1=18 5 0 1 >5 5 2 34 5 6 6 6 f 5 =18+0-1=17 (f 3 =18-1=17)

30. 30 2-4 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 6 0 1 >5 5 4 2 34 A0A0 A1A1 6 0 1 5 3 4 2 34 6 0 1 5 2 34 6 6 6 g 4 =12+1-1=12 5 0 1 >5 5 2 34 5 6 6 g 4 =12+0-1=11 (g 2 =12-1=11)

31. 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 … … … 0 0 0 Remove chips 2 k’ 0 1 >e>e e k-1 k 0 … … … 0 0 0 0 0 1 Repeat until 1 chip left at position e … …

32. 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 … … … 0 0 0 2 k’ 0 error fraction

33. 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