Succinct Representation of Codes with Applications to Testing

Abstract

Motivated by questions in property testing, we search for linear error-correcting codes that have the “single local orbit” property: 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 ${\mathbb{F}}_{2^n}$ for prime n, and whose symmetry group includes the group of non-singular affine transformations of ${\mathbb{F}}_{2^n}$, has the single local orbit property. (A code is sparse if it contains polynomially many codewords in its block length.) In particular this class includes the dual-BCH codes for whose duals (BCH codes) simple bases were not known. Our result gives the first short (O(n)-bit, as opposed to $\exp(n)$-bit) description of a low-weight basis for BCH codes. If 2ⁿ − 1 is a Mersenne prime, then we get that every sparse cyclic code also has the single local orbit.