Relaxed Locally Correctable Codes with Improved Parameters Vahid

Relaxed Locally Correctable Codes with Improved Parameters Vahid R. Asadi Joint work with Igor Shinkar Simon Fraser University NOVEMBER 2020 1

Error-correcting Codes 2

Locally Decodable Codes (LDCs) Local Decoder Corrupted Codeword 3

An example: Hadamard Code 4

Locally Correctable Codes (LCCs) Local Decoder Corrupted Codeword 5

LDC Properties The good: Many application! For instance, in cryptography, complexity, construction of PCPs, and … The bad: Poor rate! The best known constructions have superpolynomial length [Yek 08], [Efr 12]. The ugly: Huge gap between constructions and lower bounds! Best known lower bounds are super-linear [KT 00], [Woo 07]. 6

Relaxed Locally Decodable Codes (RLDCs) Introduced by Ben-Sasson et al. [BGHSV 06] Key idea: Allow the decoder to say “Don’t know” if it detects that the input is noisy. This suffices for many applications. 7

Relaxed Locally Decodable Codes Relaxed Local Decoder Corrupted Codeword 8

Relaxed Locally Correctable Codes Relaxed Local Decoder Corrupted Codeword 9

Main Result 10

About Construction 11

Reed-Muller Codes 12

How to check for errors? • An important property of Reed-Muller codes that we’ll use: • • Restriction of Reed-Muller codes to planes and lines are also low-degree polynomials. In other words, they’re codewords of smaller dimension. • So, we can use the planes and lines to do some local tests. 13

Local Tests 14

Everything looks good, but… 15

PCPPs 16

c. PCPPs 17

Some Issues… 18

Consistency Tests We use two type of consistency tests: • Plane vs. point: check whether a point is consistent with the rest of the plane. • Plane vs. line: check whether a line is consistent with the rest of the plane 19

Plane vs Point Test 20

Plane vs Line Test 21

Consistency Test using Random Walk • Defined in [CGS 20] for tensored codes. • Here we modify it for Reed-Muller codes. • How the random walk works? • • Start by picking a plane which contains the desired point. At each step, sample a line from previous plane, then sample another plane that contains that line. Do the above iteration until you reach a uniform plane. Check whether all the planes are low-degree. 22

So far… • We have found a solution for the first issue. • We still need to reduce the number of planes we use. • Need two general properties from the planes: • • Total number of planes is not very much! Rapid mixing in the random walk. 23

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Relaxed Local Decoding Algorithm 26

CTRW Theorem 27

Code Construction 28

Composition Theorem 29

What about RLCC? The idea to make this code an RLCC is to encode the proofs with some systematic RLCC (i. e. , [CGS 20], [GRR 18]). • So you can access the proof from the message part of the code. • Also, you are able to decode locally decode the proof, using the inner RLCC. • Be careful! First, you need to make sure that the c. PCPP proof actually corresponds to the lowdegree test. 30

Open Problems 31

Thanks! Stay safe, and stay relaxed! 32