Locally testable cyclic codes

1. Locally testable cyclic codes László Babai, Amir Shpilka, Daniel Štefankovič Are there good cyclic codes ??? [Open Problem 9.2 in MacWilliams, Sloane ’77] Are there good locally-testable codes ??? [Goldreich, Sudan ’02] Theorem: There are no good families of locally-testablecyclic codes over . This talk

2. Linear codes – basic parameters code = linear subspace of alphabet size block size dimension = information length minimum weight = distance

3. A good family of codes linear information (const rate) linear distance ( errors corrected) existence [Shannon’48]. explicit [Justesen’72].

4. Cyclic codes [Prange’57] 0 1 1 0 0 1 0 1 1 0 1 1 0 0 1 0 Why cyclic codes? Hardware – shift registers Classical codes – BCH, Reed-Solomon Theory – principal ideal rings [MacWilliams,Sloane’77] ”Cyclic codes are the most studied of all codes since they are easy to encode and include the important family of BCH codes.”

5. Are there good cyclic codes? ??? [Lin, Weldon ’67] BCH codes are not good [Berman ’67] If the largest prime divisor of is then the family cannot be good. [BSS ’03] If the largest prime divisor of is then the family cannot be good.

6. Local testability (context: holographic proofs/PCPs) a word randomized tester check few bits - randomized codeword - surely accepted far from all codewords - likely rejected

7. Holographic proofs/PCPs [Babai, Fortnow, Lund ‘91] polylog bits checked, quasipoly length [Babai, Fortnow, Levin, Szegedy ’91, 20??] polylog bits checked, nearly linear length [Arora, Lund, Motwani, Sudan, Szegedy’92] const bits checked, polynomial length Locally testable codes [Friedl, Sudan’95] – local testability formalized const bits checked, nearly quadratic length [Goldreich, Sudan’02] const bits checked, nearly linear length Clarified PCP loc. testable codes connection

8. Locally decodable codes strengthening of local testability stronger tradeoffs known [Katz, Trevisan ’00] [Goldreich, Karloff, Schulman, Trevisan ‘02] [Deshpande, Jain, Kavitha, Lokam, Radhakrishnan ’02] [Kerenidis, de Wolf ’03]

9. Are there good locally testable cyclic codes? is smooth contains a large prime no good cyclic code no good locally testable cyclic code [Berman ’67, BSS ‘03] No local testability assumption needed!

10. Our lower bound proof works against • adaptive tester • codeword always accepted • word at distance rejected with • positive probability TRADEOFF: If L bits tested then either information length or distance

11. Idea of proof – illustrated CASE: prime + cyclic pattern tester word randomized tester accept iff fixed uniformly random from Method of proof: Diophantine approximation

12. Dirichlet’s Theorem (simultaneous Diophantine approximation) For any integer , reals it is possible to simultaneously approximate by rationals with error bounded by :

13. prime + cyclic pattern tester word “spread” of the tester: shortest arc which includes an instance of the pattern determines the codeword “spread” dimension spread-1

14. The trick: We shrink the spread to without changing the dimension. Corollary: dimension code not good Q.E.D.

15. Q: How to shrink the spread? A: Stretch the code . Stretch factor New code: cyclic, same dimension We can even use our old tester! Instead of querying positions , query positions

16. Lemma: If is prime then there exists a stretch factor which reduces the spread to Proof: apply Dirichlet’s Theorem to approximating with denominator The stretch factor will be the common denominator.

17. Algebraic machinery for cyclic codes cyclicity : check polynomial of cyclic code Information length

18. We need to understand divisors of degree of over . Factoring over cyclotomic polynomial of order s

19. very sparse, weight independent of irreducible over but not over (ignore for now)

20. If is smooth (all prime divisors small) then large , small , such that We use the sparsity of to show codeword small weight So, code not good. Q.E.D?

21. irreducible over but not over (don’t ignore) very sparse, weight independent of even the irreducible factors exhibit similar pattern of sparsity So, code not good. Q.E.D! (“some” technical details omitted :-)

22. Are there good cyclic codes? Mersenne prime have degree Factors of in Conjecture: Random cyclic code with Mersenne prime block length is good.