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 be 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

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... eption was in the work of Ben-Sasson et al. [SIAM J. Computing, 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. [IEEE Trans. Inf. Th., 2005]. If true such a result would capture many of the basic ingredients of known locally testable codes.