Motivated by questions in property testing, we 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_{2^n} for prime n, and whose
symmetry group includes the group of non-singular affine transformations
of F_{2^n}, has the single local orbit property. (A code is said to be
{\em 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_{2^n}
for prime n are locally testable.

If, in addition to n being prime, if 2^n-1 is also prime
(i.e., 2^n-1 is a Mersenne prime),
then we get that every sparse {\em cyclic} code also has the single
local orbit. In particular this implies that BCH codes of Mersenne
prime length are generated by a single low-weight codeword and
its cyclic shifts.

In retrospect, the single local orbit property has been central
to most previous results in algebraic property testing.
However, in the previous cases, the single local property was
almost ``evident'' for the code in question (the single local
constraint was explicitly known, and it is a simple
algebraic exercise to show that its translations under the symmetry
group completely characterize the code). Our work gives an alternate
proof of the single local orbit property, effectively by counting,
and its effectiveness is demonstrated by the fact that we are
able to analyze it in cases where even the local constraint is
not ``explicitly'' known.
Our techniques involve the use of recent results from additive number
theory to prove that the codes we consider, and related codes emerging
from our proofs, have high distance. We then combine these with the
MacWilliams identities and some careful analysis of the
invariance properties to derive our results.