Locally testable cyclic codes

Cyclic linear codes of block length n over a finite field F_q are linear subspaces of F_qⁿ that are invariant under a cyclic shift of their coordinates. A family of codes is good if all the codes in the family have constant rate and constant normalized distance (distance divided by block length). It is a long-standing open problem whether there exists a good family of cyclic linear codes. A code C is r-testable if there exists a randomized algorithm which, given a word x ∈ F_qⁿ, adaptively selects r positions, checks the entries of x in the selected positions, and makes a decision (accept or reject x) based on the positions selected and the numbers found, such that i) if x ∈ C then x is surely accepted; ii) if dist(x, C) ≥ ∈n then x is probably rejected. ("dist" refers to Hamming distance.) A family of codes is locally testable if all members of the family are r-testable for some constant r. This concept arose from holographic proofs/PCP's. Recently it was asked whether there exist good, locally testable families of codes. In this paper the intersection of the two questions stated is addressed. Theorem. There are no good, locally testable families of cyclic codes over any (fixed) finite field. In fact the result is stronger in that it replaces condition ii) of local testability by the condition ii') if dist(x, C) ≥ ∈n then x has a positive chance of being rejected. The proof involves methods from Galois theory, cyclotomy, and diophantine approximation.