Untitled

Ben–Sasson, Eli; Viderman, Michael

doi:10.4086/toc.2009.v005a012

Cited by 27 publications

(2 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This notion has also found applications to proof composition in the setting of PCPs [10]. These articles have motivated the study of the robustness of the axis-parallel hyperplane test for tensor product codes, establishing both positive results [22,14,15,35] and limitations [34,20,24].…”

Section: Ltc Constructionsmentioning

confidence: 99%

Untitled

Chiesa¹,

Manohar²,

Shinkar³

2020

Theory of Comput.

View full text Add to dashboard Cite

Many low-degree tests examine the input function via its restrictions to random hyperplanes of a certain dimension. Examples include the line-vs-line (Arora, Sudan 2003), plane-vs-plane (Raz, Safra 1997), and cube-vs-cube (Bhangale, Dinur, Navon 2017) tests. In this paper we study tests that only consider restrictions along axis-parallel hyperplanes, which have been studied by Polishchuk and Spielman (1994) and Ben-Sasson and Sudan (2006). While such tests are necessarily "weaker," they work for a more general class of codes, namely tensor product codes. Moreover, axis-parallel tests play a key role in constructing LTCs with inverse polylogarithmic rate and short PCPs (Polishchuk, Spielman 1994; Ben-Sasson, Sudan 2008; Meir 2010). We present two results on axis-parallel tests. (1) Bivariate low-degree testing with low agreement. We prove an analogue of the Bivariate Low-Degree Testing Theorem of Polishchuk and Spielman in the low-agreement regime, albeit for much larger fields. Namely, for the tensor product of the Reed-Solomon A preliminary version of this paper appeared in the Proceedings of the 21st International Workshop on Randomization and Computation (RANDOM'17). * Supported by the Center for Long-Term Cybersecurity at UC Berkeley.

show abstract

Section: Ltc Constructionsmentioning

confidence: 99%

Untitled

Chiesa¹,

Manohar²,

Shinkar³

2020

Theory of Comput.

View full text Add to dashboard Cite

show abstract

“…Unique neighbour expansion is also important for error-correcting codes with additional features. In [12,13], it is shown that codes constructed from unique neighbour expanders are weakly smooth and can be used to construct robustly testable codes. This property, in turn, is important for the composition of locally testable codes, and, more recently, for quantum LDPC codes (see [14,15]).…”

Section: Related Workmentioning

confidence: 99%

Bipartite Unique Neighbour Expanders via Ramanujan Graphs

Asherov,

Dinur

2024

Entropy

View full text Add to dashboard Cite

We construct an infinite family of bounded-degree bipartite unique neighbour expander graphs with arbitrarily unbalanced sides. Although weaker than the lossless expanders constructed byCapalbo et al.,our construction is simpler and may be closer to being implementable in practice, due to the smaller constants. We construct these graphs by composing bipartite Ramanujan graphs with a fixed-size gadget in a way that generalises the construction of unique neighbour expanders by Alon and Capalbo. For the analysis of our construction, we prove a strong upper bound on average degrees in small induced subgraphs of bipartite Ramanujan graphs. Our bound generalises Kahale’s average degree bound to bipartite Ramanujan graphs, and may be of independent interest. Surprisingly, our bound strongly relies on the exact Ramanujan-ness of the graph and is not known to hold for nearly-Ramanujan graphs.

show abstract

A combination of testability and decodability by tensor products

Viderman

2013

Random Struct. Alg.

Self Cite

View full text Add to dashboard Cite

Ben‐Sasson and Sudan (RSA 2006) showed that taking the repeated tensor product of linear codes with very large distance results in codes that are locally testable. Due to the large distance requirement the associated tensor products could be applied only over sufficiently large fields. Then Meir (SICOMP 2009) used this result to present a combinatorial construction of locally testable codes with largest known rate. As a consequence, this construction was obtained over sufficiently large fields. In this paper we improve the result of Ben‐Sasson and Sudan and show that for any linear codes the associated tensor products are locally testable. Consequently, the construction of Meir can be taken over any field, including the binary field. Moreover, a combination of our result with the result of Spielman (IEEE IT, 1996) implies a construction of linear codes (over any field) that combine the following properties: have constant rate and constant relative distance; have blocklength n and are testable with nϵ queries, for any constant ϵ > 0; linear time encodable and linear‐time decodable from a constant fraction of errors. Furthermore, a combination of our result with the result of Guruswami et al. (STOC 2009) implies a similar corollary for list‐decodable codes. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 46, 572–598, 2015

show abstract

Untitled

Cited by 27 publications

References 14 publications

Untitled

Untitled

Bipartite Unique Neighbour Expanders via Ramanujan Graphs

A combination of testability and decodability by tensor products

Contact Info

Product

Resources

About