Locally testable code: Difference between revisions

Revision as of 22:06, 5 December 2021

Revision as of 22:33, 5 December 2021

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== Limits ==

It remains an open question whether there are any locally testable codes of linear size, but there are several constructions that are considered "nearly linear":<ref name=shortLTC>{{cite web|url=http://eccc.hpi-web.de/eccc-reports/2005/TR05-014/index.html|title=Short Locally Testable Codes and Proofs (Survey)|first=Oded|last=Goldreich|year=2005|citeseerx = 10.1.1.110.2530}}</ref>

# Polynomial arbitrarily close to linear; for any <math>\epsilon>0</math>, <math>n=k^{1+\epsilon}</math>.

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The next nearly linear goal is linear up to a [[polylogarithmic]] factor; <math>n=\text{poly}(\log k)*k</math>. Nobody has yet to come up with a linearly testable code that satisfies this constraint.<ref name=shortLTC/>

In November 2021 two preprints have reported<ref>{{cite arxiv|lastlast1=Panteleev|firstfirst1=Pavel|last2=Kalachev|first2=Gleb|date=2021-11-05|title=Asymptotically Good Quantum and Locally Testable Classical LDPC Codes|arxivclass=cs.IT|eprint=2111.03654}}</ref><ref>{{cite arxiv|lastlast1=Dinur|firstfirst1=Irit|author-link=Irit Dinur|last2=Evra|first2=Shai|author-link2=Shai Evra|last3=Livne|first3=Ron|last4=Lubotzky|first4=Alexander|author-link4=Alexander Lubotzky|last5=Mozes|first5=Shahar|author-link5=Shahar Mozes|date=2021-11-08|title=Locally Testable Codes with constant rate, distance, and locality|arxivclass=cs.IT|eprint=2111.04808}}</ref><ref>{{Cite web|last=Rorvig|first=Mordechai|date=2021-11-24|title=Researchers Defeat Randomness to Create Ideal Code|url=https://www.quantamagazine.org/researchers-defeat-randomness-to-create-ideal-code-20211124/|url-status=live|access-date=2021-11-24|website=[[Quanta Magazine]]|language=en}}</ref> the first polynomial-time construction of "<math>c^3</math>-LTCs" namely locally testable codes with constant rate <math>r</math>, constant distance <math>\delta</math> and constant locality <math>q</math>.

== Connection with probabilistically checkable proofs ==