Locally Testable Cyclic Codes

Laszlo Babai; Amir Shpilka; Daniel Stefankovic. 12 August, 2003.
Communicated by Laszlo Babai.


Cyclic linear codes of block length n over a finite field F_q are linear subspaces of 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 exists a randomized algorithm which, given a word x in 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;

(ii) if dist(x,C) >= epsilon*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/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.

Theorem. There are no good, locally testable families of cyclic codes over any (fixed) finite field.

In fact our result is stronger in that it replaces condition (ii) of local testability by the condition

(ii') if dist(x,C) >= epsilon*n then x has a positive chance of being rejected.

The proof involves methods from Galois theory, cyclotomy, and diophantine approximation.

Original Document

The original document is available in PDF (uploaded 12 August, 2003 by Laszlo Babai).