Limitations on Testable Affine-Invariant Codes in the High-Rate Regime

Locally testable codes (LTCs) of constant minimum (absolute) distance that allow the tester to make a nearly linear number of queries have become the focus of attention recently, due to their connections to central questions in approximability theory. In particular, the binary Reed-Muller code of block length N and absolute distance d is known to be testable with O(N/d) queries, and has a dimension of ≈ N − (logN) . The polylogarithmically small co-dimension is the basis of constructions of small set expanders with many “bad” eigenvalues, and size-efficient PCPs based on a shorter version of the long code. The smallest possible codimension for a distance d code (without any testability requirement) is ≈ d2 logN , achieved by BCH codes. This raises the natural question of understanding where in the spectrum between the two classical families, Reed-Muller and BCH, the optimal co-dimension of a distance d LTC lies — in other words the “price” one has to pay for local testability. One promising approach for constructing LTCs is to focus on affine-invariant codes, whose structure makes testing guarantees easier to deduce than for general codes. Along these lines, the authors of [HRZS13] and [GKS13] recently constructed an affine-invariant family of high-rate LTCs with slightly smaller co-dimension than Reed-Muller codes. In this work, we show that their construction is essentially optimal among linear affine-invariant LTCs that contain the Reed-Muller code of the appropriate degree. ∗Carnegie Mellon University, [email protected]. Part of this work was done while visiting Microsoft Research New England. Research supported in part by NSF grant CCF-0963975. †Microsoft Research New England, [email protected]. ‡Carnegie Mellon University, [email protected]. Part of this work was done while visiting Microsoft Research New England. Research supported by NSF grant CCF-0963975 and MSR-CMU Center for Computational Thinking. §Carnegie Mellon University, [email protected]. Research supported by NSF grant CCF-0963975 and NSF Graduate Fellowship.