Name
Abstract | ||
|---|---|---|
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. |
Year | DOI | Venue |
|---|---|---|
2011 | 10.1109/CCC.2011.14 | CCC '11 Proceedings of the 2011 IEEE 26th Annual Conference on Computational Complexity |
Keywords | Field | DocType |
parity check codes,alon,ben-sasson,affine-invariant codes,complexity theory,locally testable codes,low density parity check codes,symmetric ldpc codes,error-correcting codes,invariance,property testing,ldpc code | Forward error correction,Discrete mathematics,Error floor,Low-density parity-check code,Reed–Muller code,Mathematics | Conference |
ISSN | ISBN | Citations |
1093-0159 E-ISBN : 978-0-7695-4411-3 | 978-0-7695-4411-3 | 12 |
PageRank | References | Authors |
0.53 | 12 | 4 |
Name | Order | Citations | PageRank |
|---|---|---|---|
| Eli Ben-Sasson | 1 | 1641 | 86.98 |
| Ghid Maatouk | 2 | 24 | 1.53 |
| Amir Shpilka | 3 | 1095 | 64.27 |
| Madhu Sudan | 4 | 5616 | 591.68 |