| Summary: | 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[subscript 2n] for prime n, and whose symmetry
group includes the group of non-singular affine transformations of F[subscript 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 F2n for prime n are
locally testable. If, in addition to n being prime, if 2n−1 is also prime (i.e., 2n−1 is a Mersenne prime), then
we get that every sparse 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.
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