Succinct representation of codes with applications to testing

Motivated by questions in property testing, w e search for linear error-correcting codes that have the "single local orbit" property, i.e., they are specified by a single local constraint and its translations under the symmetry group of the code. We show that the dual of every "sparse" binary code whose coordinates are indexed by elements of F_2n for prime n and whose symmetry group includes the group of nonsingular affine transformations of F_2n has the single local orbit property. (A code is said to be sparse if it contains polynomially many codewords in its block length.) In particular this class includes the dual-BCH codes for whose duals (i.e., for BCH codes) simple bases were not known. Our result gives the first short (O(n)-bit, as opposed to the natural exp(n)-bit) description of a low-weight basis for BCH codes. The interest in the single local orbit property comes from the recent result of Kaufman and Sudan (STOC 2008) that shows that the duals of codes that have the single local orbit property under the affine symmetry group are locally testable. When combined with our main result, this shows that all sparse affine-invariant codes over the coordinates F_2n for prime n are locally testable. If, in addition to n being prime, 2ⁿ - 1 does not have large divisors, then we get that every sparse cyclic-invariant code also has the single local orbit. In particular this implies that BCH codes of such length are generated by a single low-weight codeword and its cyclic shifts.