Succinct representation of codes with applications to testing

1. Succinct representation of codes with applications to testing Elena Grigorescu Tali Kaufman Madhu Sudan

2. Outline • Testing membership in error correcting codes • Sufficient conditions for testing algebraic codes • Possible promising perspective: rich group of symmetries of code • Our result: affine/cyclic invariant, sparse codes can be described succinctly by a single, short codeword • Implies locally testability results • Proof sketch • Conclusions

3. Locally testable codes qqueries C C -Acceptw.p 1 if -Reject w.p. ε if ( independent of n) C satisfies Code: Linear:

4. Testing linear codes via duality • [BHR] Test for linear properties are essentially of the form: • Given x, pick • Accept iff • Locality of test: • Dual-distance:smallest weight of a codeword in dual-C

5. Sufficient conditions for testing • Necessary condition for local testing (linear codes): - small “dual distance” - not sufficient( [BHR] show random LDPC not locally testable) • Sufficient conditions - Possible approach: nice symmetries of code • C is invariant under permutation iff

6. Symmetries and testing • Many known testable codes have somewhat large symmetry groups: Eg. Linearity: invariance under general linear group Low degree, Reed-Muller, BCH: invariance under affine group • Specific sufficient condition: [KS] affine invariance + ‘local characterization’ imply testing • AKKLR Conjecture: 2 transitivity + small dual distance Falsified in general [GKS] • Modified AKKLR Question: What if dual code is generated by single low-weight codeword and its shifts under some group G (“Single-Orbit Property under G”) Are these codes testable (for some group G? for all groups G?)

7. Single orbit property under affine invariant/cyclic groups • Affine group: • Cyclic group: • C has single orbit under cyclic group: w=01001 then B={01001, 10100, 01010, 00101, 10010} is a basis for C • Formally, C has k-single orbit under G ( included in Aut(C) ) if

8. Our work • Study “Single-Orbit Property” of common codes. • Def: C is sparse if it contains a poly number of codewords • Duals of binary sparse + affine invariant codes have the single-orbit property under affine group - under some block-length restriction: n prime - [KS’08] Single-orbit codes under affine group are testable. • Duals of binary sparse + cyclic invariant codes have the single-orbit property under cyclic group - under more block-length restrictions: n, N-1 primes - No testing implications

9. Related works • Sparse, large distance codes are testable [KL, KS] ( tests are coarse, unstructured) • Affine/linear invariant + “characterization” imply testing • Here: sparselarge distance affine invariance“characterization” (explicit tests) • [KL] dual-e-BCH codes are testable (unstructured tests) • e-BCH are spanned by shortest codewords • Here: dual-e-BCH are spanned by a single, short codeword (explicit basis / tests)

10. Toward an explicit description of binary affine invariant codes Affine invariance: Any function is of the form The Tracefunction:

11. Explicit description of sparse affine families • Let - What aff inv families does f belong to? • Consider the binary rep of degrees: 1, 111, 1100, 10011 • Then • In general: if degree d occurs then its shadow occurs • Sparsity translates into few monomials • Affine/Cyclic codes are described by a small set of degrees Shadow(10011) = {10011,10010,10001,10000,11,10,1}

12. Proof ingredients • Strong number theoretic result of Bourgain implies high weight of functions of the form few degs > deg< Degs inside trace 0 ? Weil bounds Bourgain

13. Proof ingredients (contd) • MacWilliams type counting estimates - fourier transform between the functions that represent number of codewords for each weight in C and in dual- C, respectively • For sparse codes of length N and of high distance obtain:

14. C described by set of degrees D Let dual-C’= Span( aff(w) ) If C’ C then there exists Let Associate C(a) to codew. w Does every wt<k codew. belong to a dual of some C(a) ? New goal: exists w that does not belong to the dual of any C(a), for all a We show weight<k Proof sketch Want: exists codew. c with wt < k s.t. Span(aff(c))=Dual-C Dual-C C’ C(a) C w Dual-C’

15. Proof Sketch • C, C(a):sparse, high dist (Bourgain) (assuming N-1 and n are primes) • How many codew of wt k in dual-C? • How many codew of wt k in dual-C(a) ? • Total number of degrees a to consider: N/n • Therefore, there exists codew. of wt<k in dual-C that whose orbit generates C

16. Specifics of the affine case proof • Here only assume n prime- Bourgain doesn’t hold for all monomials • Need codes C(a) to have deg a < • Use shadow property • Show that enough to consider a in the set

17. Cyclic codes • Invariant under: • Punctured affine invariant codes are cyclic • Cyclic codes are described by generator polynomial (or its roots in the field) • Alternatively described by function families of the form • Degrees can be arbitrary

18. Affine (length N= ) n prime degrees of monomials are shadow closed |Aut(C)|= “single orbit” implies testing Cyclic (length N-1) n, N-1 primes degrees of monomials are arbitrary |Aut(C)|=N not known if “single orbit” implies testing Single orbit: affine vs cyclic codes

19. Open Questions • Do same results hold for non-prime n, ? • Single orbit under what other groups imply testing? How large does the Aut group should be to imply testing? • Small weight basis + invariance implies testing? • Examples of families where the tests are not the “expected” ones (I.e. not the ones suggested by the description of Aut group)

20. Thank you