TY - JOUR

T1 - Testing Reed-Muller codes

AU - Alon, Noga

AU - Kaufman, Tali

AU - Krivelevich, Michael

AU - Litsyn, Simon

AU - Ron, Dana

PY - 2005/11

Y1 - 2005/11

N2 - A code is locally testable if there is a way to indicate with high probability that a vector is far enough from any codeword by accessing only a very small number of the vector's bits. We show that the Reed-Muller codes of constant order are locally testable. Specifically, we describe an efficient randomized algorithm to test if a given vector of length n = 2m is a word in the rth-order Reed-Muller code R(r, m) of length n = 2m. For a given integer r ≥ 1, and real ∈ > 0, the algorithm queries the input vector v at O(1/∈ + r22r) positions. On the one hand, if v is at distance at least ∈n from the closest codeword, then the algorithm discovers it with probability at least 2/3. On the other hand, if v is a codeword, then it always passes the test. Our result is almost tight: any algorithm for testing R(r, m) must perform Ω(1/∈ + 2r) queries.

AB - A code is locally testable if there is a way to indicate with high probability that a vector is far enough from any codeword by accessing only a very small number of the vector's bits. We show that the Reed-Muller codes of constant order are locally testable. Specifically, we describe an efficient randomized algorithm to test if a given vector of length n = 2m is a word in the rth-order Reed-Muller code R(r, m) of length n = 2m. For a given integer r ≥ 1, and real ∈ > 0, the algorithm queries the input vector v at O(1/∈ + r22r) positions. On the one hand, if v is at distance at least ∈n from the closest codeword, then the algorithm discovers it with probability at least 2/3. On the other hand, if v is a codeword, then it always passes the test. Our result is almost tight: any algorithm for testing R(r, m) must perform Ω(1/∈ + 2r) queries.

KW - Affine subspaces

KW - Binary field

KW - Multivariate polynomials

KW - Property testing

KW - Reed-Muller code

UR - http://www.scopus.com/inward/record.url?scp=27744594919&partnerID=8YFLogxK

U2 - 10.1109/TIT.2005.856958

DO - 10.1109/TIT.2005.856958

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AN - SCOPUS:27744594919

SN - 0018-9448

VL - 51

SP - 4032

EP - 4039

JO - IEEE Transactions on Information Theory

JF - IEEE Transactions on Information Theory

IS - 11

ER -