Resilience of Polynomials

1. Resilience of Polynomials MadhuSudan Harvard IBMQ - Resilience of Polynomials

2. Outline of talk IBMQ - Resilience of Polynomials • Motivation: Reliable Communication • Reed-Solomon Codes = Codes from Polynomials • Decoding Reed-Solomon Codes: • Welch-Berlekamp ‘86 (see Gemmell-S. ‘92) • List-decoder via Bivariate interpolation (S. ‘96) • Mutliplicity decoding (Guruswami-S’98) • Updates + Conclusions

3. Reliable Communication? We are not ready We are now ready Alice Bob IBMQ - Resilience of Polynomials • Problem from the 1940s: Advent of digital age. • Communication media are always noisy • But digital information less tolerant to noise!

4. Coding by Repetition IBMQ - Resilience of Polynomials • Can repeat (every letter of) message to improve reliability: WWW EEE AAA RRR EEE NNN OOO WWW … WXW EEA ARA SSR EEE NMN OOP WWW … • Calculations: • repetitions Prob. Single symbol corrupted • To transmit symbols, choose • Rate of transmission as • Belief (pre-1940s): Rate of any scheme as

5. Shannon’s Theory [1948] Alice Encoder Bob Decoder IBMQ - Resilience of Polynomials • Sender “Encodes” before transmitting • Receiver “Decodes” after receiving • Encoder/Decoder arbitrary functions. • Rate = • Requirement: w. high prob. • What are the best (with highest Rate)?

6. Shannon’s Theorem IBMQ - Resilience of Polynomials • If every bit is flipped with probability • Rate can be achieved. • This is best possible. • Examples: • Monotone decreasing for • Positive rate for ; even if

7. Challenges post-Shannon IBMQ - Resilience of Polynomials • Encoding/Decoding functions not “constructive”. • Shannon picked at random, brute force. • Consequence: • takes time to compute (on a computer). • takes time to find! • Algorithmic challenge: • Find more explicitly. • Both should take time to compute

8. II. Codes from Polynomials IBMQ - Resilience of Polynomials

9. Explicit Codes: Reed-Solomon Code IBMQ - Resilience of Polynomials • Messages = Coefficients of Polynomials. • Example: • Message = • Think of polynomial • Encoding: • First four values suffice, rest is redundancy! • (Easy) Facts: • Any values sufficewhere = length of message. • Can handle erasures or errors. • Explicit encoding  • Efficient decoding? [Peterson 1960]

10. Formally: IBMQ - Resilience of Polynomials • Reed-Solomon Codes:[RS ‘60] • Spec. by distinct • Message: ; • Encoding: • Parameters: • Length = ; Dimension = ; Distance • Optimal [Singleton ’64] • Algorithmics: • Encoding: Poly Evaluation (Linear Algebra) • Error-detection: Interpolation (Linear Algebra) • Error-correction? Non-trivial!

11. III. Error-correction (“Resilience of Polynomials”) IBMQ - Resilience of Polynomials

12. Error-correction Problem IBMQ - Resilience of Polynomials • Given: , • Task: Find s.t. • Naïve solutions: • Enumerate codewords: • Enumerate error locations: • Efficient Solutions: • [Peterson ‘60]: • [Berlekamp+Massey ’70]: • [Welch-Berlekamp ‘92]: Elegant explanation!

13. III(a) - Welch-Berlekamp IBMQ - Resilience of Polynomials • Key notion: Error-locator Polynomial • ; • Some properties: • Algorithm: Ignore above, and find ! • Find s.t • Output if it is a polynomial. • Just a linear system!

14. Analysis IBMQ - Resilience of Polynomials • satisfying our requirements exists! • Use error locator; • (Doesn’t imply algorithm will find this pair!) • If algorithm finds then ! • & • QED! • Theorem: If then errors can be corrected in time

15. III(b) - More Errors? List Decoding IBMQ - Resilience of Polynomials • Why was the limit for #errors? • is clearly a limit – right? • First half = evaluations of • Second half = evaluations of • What is the right message: or ? • (even ) is the limit for “unique” answer. • List-decoding: Generalized notion of decoding. • Report (small) list of possible messages. • Decoding “successful” if list contains the message polynomial.

16. Reed-Solomon List-Decoding Problem IBMQ - Resilience of Polynomials • Given: • Parameters: • Points: in the plane (finite field actually) • Find: • All degree poly’s that pass thru of points • i.e., all s.t. • : Answer unique; [Peterson 60, WB’86] • [S. 96, Guruswami+S. ‘98]: ; small list

17. Decoding by example + picture [S’96] IBMQ - Resilience of Polynomials Algorithm idea: Find algebraic explanation of all points. Stare at the solution  (factor the polynomial) Ssss

18. Decoding by example + picture [S’96] IBMQ - Resilience of Polynomials Algorithm idea: Find algebraic explanation of all points. Stare at the solution  (factor the polynomial) Ssss

19. Decoding Algorithm • Fact: There is always a degree polynomial thru points • Can be found in polynomial time (solving linear system). • [80s]: Polynomials can be factored in polynomial time [Grigoriev, Kaltofen, Lenstra] • Leads to (simple, efficient) list-decoding correcting fraction errors for IBMQ - Resilience of Polynomials

20. III(c) Even more errors? [Guruwami-S’98] IBMQ - Resilience of Polynomials Fit deg curve? Reason: at least 5 lines with 4 points each Typical point on 2 lines! New idea: Find deg7 curve passing thru each point twice! (margin too small)

21. What happened? IBMQ - Resilience of Polynomials • Condition that every point is a zero twice leads to 3 linear constraints per point. Total of 33. • polynomial satisfying 33 constraints? • Yes - #coeffs • # common zeroes (of line and ) • (each intersection point counts twice!) • Letting multiplicities , get algorithm correcting from fraction errors! • Error fraction strictly better than for every

22. Subsequent developments IBMQ - Resilience of Polynomials • [Parvaresh-Vardy’05],[Guruswami-Rudra’06]: “Folded Reed-Solomon Codes” • Correct fraction errors! • [Kopparty-Saraf-Yekhanin ’10]: Multiplicity Codes: • Encode by evaluating polynomial and its derivatives! • Nice locality features! • [Dvir’06],[Dvir-Kopparty-Saraf-S’10]: Bounds on size of Kakeya sets by Polynomial method and method of multiplicities.

23. Summary and conclusions IBMQ - Resilience of Polynomials • (Many) errors can be dealt with: • Pre-Shannon: vanishing fraction of errors • Pre-list-decoding: small constant fraction • Post-list-decoding: overwhelming fraction • Future challenges? • Communication can overcome errors introduced by channels. • Can communication overcome errors in misunderstanding between sender and receiver? • [Goldreich,Juba,S. ‘2011]; [Juba,Kalai,Khanna,S.’2011] ….

24. Conclusions IBMQ - Resilience of Polynomials • Simple algebra leads to • Amazing codes • Surprising algorithms • Even striking combinatorics • Power of linearization! • Ignore you have factorization. • You will get it anyway! • Only algorithm is for linear systems. • Finite fields vs. Reals/Complexes: • Algorithm works fine over finite fields • But over reals/complexes – algorithm fails • Even problem is not stable. New challenges ahead!

25. Thank You! IBMQ - Resilience of Polynomials