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Symmetric LDPC codes are not necessarily locally testable

By Eli Ben-sasson, Ghid Maatouk, Amir Shpilka and Madhu Sudan


Locally testable codes, i.e., codes where membership in the code is testable with a constant number of queries, have played a central role in complexity theory. It is well known that a code must be a “low-density parity check ” (LDPC) code for it to be locally testable, but few LDPC codes are known to the locally testable, and even fewer classes of LDPC codes are known not to be locally testable. Indeed, most previous examples of codes that are not locally testable were also not LDPC. The only exception was in the work of Ben-Sasson et al. [2005] who showed that random LDPC codes are not locally testable. Random codes lack “structure ” and in particular “symmetries ” motivating the possibility that “symmetric LDPC ” codes are locally testable, a question raised in the work of Alon et al. [2005]. If true such a result would capture many of the basic ingredients of known locally testable codes. In this work we rule out such a possibility by giving a highly symmetric (“2-transitive”) family of LDPC codes that are not testable with constant number of queries. We do so by continuing the exploration of “affine-invariant codes ” — codes where the coordinates of the words are associated with a finite field, and the code is invariant under affine transformation

Topics: Property testing, Invariance, Error-correcting codes
Year: 2010
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