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 (cf. [MS. p.270]). 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) ≥ εn 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/PCPs. Goldreich and Sudan [GS] asked whether there exist good, locally testable families of codes. In this paper we address the intersection of the two questions stated.