Abstract
Cyclic linear codes of block length n over a finite field \mathbb{F}_q are the linear subspace of \mathbb{F}_{_q }^n 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 exist a randomized algorithm which, given a word x \in \mathbb{F}_q^n, 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 \in C then x is surely accepted; if dist(x,C) \geqslant \varepsilon n then x is probably rejected. ("dist" refers to Hamming distane.) A family of codes is locally testable if all members of the family are r-testable for some 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.