Succinct Representation of Codes with Applications to Testing

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 F2n for prime n, and whose symmetry group includes the group of non-singular affine transformations of F2n , 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...

• Type: Conference
• Year: 2009
• Where: APPROX
• Authors: Elena Grigorescu, Tali Kaufman, Madhu Sudan
• Keywords: Algorithms, APPROX 2009, Local Orbit Property, Single Local Orbit, Symmetry Group