Breaking the ε-Soundness Bound of the Linearity Test over GF(2)
APPROX '08 / RANDOM '08 Proceedings of the 11th international workshop, APPROX 2008, and 12th international workshop, RANDOM 2008 on Approximation, Randomization and Combinatorial Optimization: Algorithms and Techniques
On the Randomness Complexity of Property Testing
APPROX '07/RANDOM '07 Proceedings of the 10th International Workshop on Approximation and the 11th International Workshop on Randomization, and Combinatorial Optimization. Algorithms and Techniques
Breaking the $\epsilon$-Soundness Bound of the Linearity Test over GF(2)
SIAM Journal on Computing
Property testing for cyclic groups and beyond
COCOON'11 Proceedings of the 17th annual international conference on Computing and combinatorics
Property testing for cyclic groups and beyond
Journal of Combinatorial Optimization
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The main result of this paper is a near-optimal derandomization of the affine homomorphism test of Blum, Luby, and Rubinfeld [J. Comput. System Sci., 47 (1993), pp. 549-595]. We show that for any groups G and &Ggr;, and any expanding generating set S of G, the natural deramdomized version of the BLR test in which we pick an element x randomly from G and y randomly from S and test whether $f(x)\cdot f(y)=f(x\cdot y)$, performs nearly as well (depending of course on the expansion) as the original test. Moreover, we show that the underlying homomorphism can be found by the natural local “belief propagation decoding.” We note that the original BLR test uses $2\log_2 |G|$ random bits, whereas the derandomized test uses only $(1+o(1))\log_2 |G|$ random bits. This factor of 2 savings in the randomness complexity translates to a near quadratic savings in the length of the tables in the related locally testable codes (and possibly probabilistically checkable proofs which may use them). Our result is a significant generalization of recent results that either refer to the special case of the groups $G=Z_p^m$ and $&Ggr; =Z_p$ or are nonconstructive. We use simple combinatorial arguments and the transitivity of Cayley graphs (and this analysis gives optimal results up to constant factors). Previous techniques used the Fourier transform, a method which seems unextendable to general groups (and furthermore gives suboptimal bounds). Finally, we provide a polynomial time (in $|G|$) construction of a (somewhat) small ($|G|^{\epsilon}$) set of expanding generators for every group $G$, which yield efficient testers of randomness $(1+\epsilon) \log |G|$ for $G$. This result follows from a simple derandomization of a known probabilistic construction.