On sums of locally testable affine invariant properties
APPROX'11/RANDOM'11 Proceedings of the 14th international workshop and 15th international conference on Approximation, randomization, and combinatorial optimization: algorithms and techniques
Limits on the rate of locally testable affine-invariant codes
APPROX'11/RANDOM'11 Proceedings of the 14th international workshop and 15th international conference on Approximation, randomization, and combinatorial optimization: algorithms and techniques
New affine-invariant codes from lifting
Proceedings of the 4th conference on Innovations in Theoretical Computer Science
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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 exception 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. 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 a 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 transformations of the field. New to our study is the use of fields that have many subfields, and showing that such a setting allows sufficient richness to provide new obstacles to local testability, even in the presence of structure and symmetry.