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

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

C Locally testable codes C C satisfies Code: Linear: -Accept w.p 1 if -Reject w.p. ε if ( independent of n) q queries 011…0

Testing linear codes via duality ► ► [BHR] Test for linear properties are essentially of the form: 1.Given x, pick 2.Accept iff ► Locality of test: ► Dual-distance: smallest weight of a codeword in dual-C

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

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 Low degree, Reed-Muller, BCH: invariance under affine group Specific sufficient condition: [KS] affine invariance + ‘local characterization’ imply testing [KS] affine invariance + ‘local characterization’ imply testing AKKLR Conjecture: 2 transitivity + small dual distance Falsified in general [GKS] 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?) Are these codes testable (for some group G? for all groups G?)

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 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

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 - under some block-length restriction: n prime - [KS’08] Single-orbit codes under affine group are testable. - [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 - under more block-length restrictions: n, N-1 primes - No testing implications - No testing implications

► Sparse, large distance codes are testable [KL, KS] ( tests are coarse, unstructured) [KL, KS] ( tests are coarse, unstructured) ► Affine/linear invariant + “characterization” imply testing ► Here: sparse large distance affine invariance “characterization” (explicit tests) 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) Related works

Toward an explicit description of binary affine invariant codes ► Affine invariance: ► Any function is of the form ► The Trace function:

► Let - What aff inv families does f belong to? - What aff inv families does f belong to? ► Consider the binary rep of degrees: 1, 111, 1100,  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 Explicit description of sparse affine families Shadow(10011) = {10011,10010,10001,10000,11,10,1}

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

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:

Proof sketch ► 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 C weight<k Dual-C w Dual-C’ Want: exists codew. c with wt < k s.t. Span(aff(c))=Dual-C C’ C(a)

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

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

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

Single orbit: affine vs cyclic codes ► 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

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)

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