Testing of exponentially large codes, by a new extension to Weil bound for character sums

In this work we consider linear codes which are locally testable in a sublinear number of queries. We give the first general family of locally testable codes of exponential size. Previous results of this form were known only for codes of quasi-polynomial size (e.g. Reed-Muller codes). We accomplish this by showing that any affine invariant code C over Fpn of size p Ω(n) is locally testable using poly(logp |C|/n) queries. Previous general result for affine invariant codes were known only for sparse codes, i.e. codes of size p. The main new ingredients used in our proof are a new extension of the Weil bound for character sums, and a Fourier-analytic approach for estimating the weight distribution of affine invariant codes. ∗Research supported in part by a Koshland Fellowship. †Research supported by the Israel Science Foundation (grant 1300/05).