Towards a classical proof of exponential lower bound for 2-probe smooth codes

Abstract

Let C: {0,1}^n -> {0,1}^m be a code encoding an n-bit string into an m-bit string. Such a code is called a (q, c, e) smooth code if there exists a decoding algorithm which while decoding any bit of the input, makes at most q probes on the code word and the probability that it looks at any location is at most c/m. The error made by the decoding algorithm is at most e. Smooth codes were introduced by Katz and Trevisan in connection with Locally decodable codes. For 2-probe smooth codes Kerenedis and de Wolf have shown exponential in n lower bound on m in case c and e are constants. Their lower bound proof went through quantum arguments and interestingly there is no completely classical argument as yet for the same (albeit completely classical !) statement. We do not match the bounds shown by Kerenedis and de Wolf but however show the following. Let C: {0,1}^n -> {0,1}^m be a (2,c,e) smooth code and if e <= c^2/8n^2, then m >= 2^(n/320c^2 - 1)$. We hope that the arguments and techniques used in this paper extend (or are helpful in making similar other arguments), to match the bounds shown using quantum arguments. More so, hopefully they extend to show bounds for codes with greater number of probes where quantum arguments unfortunately do not yield good bounds (even for 3-probe codes).