We consider cyclic codes over a fixed finit field. A family of codes is *good* if it has constant rate and linear distance. A code is *locally testable* if for a word x it is possible to decide (with some probability of error) whether x is in the code or x is far from any codeword. It is a long-standing open problem whether there are good cyclic codes. A more recent question is whether there are good locally testable codes. We will show that there are no good cyclic, locally testable codes.
Joint work with Laszlo Babai and Amir Shpilka