Locally testable codes and expanders

A locally testable code is a code defined by a robust set of local constraints. Namely, the distance of a vector from the code is well approximated by the fraction of local constraints that it violates. A constraint graph of an LTC is a graph whose vertices are labeled by the coordinates of the code, and vertex i is adjacent to j whenever they occur together in a constraint. We study the relation between the topology of this graph and the structure of the code. We show that every constraint graph of an LTC must be a small set expander, in which all sets up to some linear size expand. Moreover, a constraint graph of an LTC can be decomposed into constantly many expander graphs on which the induced codes are approximately LTCs. Our work suggests a subtle relation between LTCs and expanders. It is known [BSHR05] that codes defined by strongly expanding (e.g. random) sets of constraints are not locally testable. In contrast, we show that every constraint graph of an LTC must be weakly expanding (i.e., small set expanders). Our result provides a necessary condition for LTCs that can be applied toward proving that certain codes are not LTCs. On the way to our result we prove that every small-set expander (i.e., a graph where small sets up to some linear size are guaranteed to expand) can be decomposed into a constant number of “standard” expanders.