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Revision #1 to TR21-151 | 16th December 2021 09:50

Locally Testable Codes with constant rate, distance, and locality

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Revision #1
Authors: Irit Dinur, Shai Evra, Ron Livne, Alexander Lubotzky, Shahar Mozes
Accepted on: 16th December 2021 09:50
Downloads: 930
Keywords: 


Abstract:

A locally testable code (LTC) is an error correcting code that has a property-tester. The tester reads $q$ bits that are randomly chosen, and rejects words with probability proportional to their distance from the code. The parameter $q$ is called the locality of the tester.

LTCs were initially studied as important components of PCPs, and since then the topic has evolved on its own. High rate LTCs could be useful in practice: before attempting to decode a received word, one can save time by first quickly testing if it is close to the code.

An outstanding open question has been whether there exist "$c^3$-LTCs", namely LTCs with *c*onstant rate, *c*onstant distance, and *c*onstant locality.

In this work we construct such codes based on a new two-dimensional complex which we call a left-right Cayley complex. This is essentially a graph which, in addition to vertices and edges, also has squares. Our codes can be viewed as a two-dimensional version of (the one-dimensional) expander codes, where the codewords are functions on the squares rather than on the edges.



Changes to previous version:

This new version corrects and polishes a few minor points, but more importantly, adds references to other related very recent works which were done independently.


Paper:

TR21-151 | 8th November 2021 22:00

Locally Testable Codes with constant rate, distance, and locality





TR21-151
Authors: Irit Dinur, Shai Evra, Ron Livne, Alexander Lubotzky, Shahar Mozes
Publication: 8th November 2021 22:01
Downloads: 1180
Keywords: 


Abstract:

A locally testable code (LTC) is an error correcting code that has a property-tester. The tester reads $q$ bits that are randomly chosen, and rejects words with probability proportional to their distance from the code. The parameter $q$ is called the locality of the tester.

LTCs were initially studied as important components of PCPs, and since then the topic has evolved on its own. High rate LTCs could be useful in practice: before attempting to decode a received word, one can save time by first quickly testing if it is close to the code.

An outstanding open question has been whether there exist "$c^3$-LTCs", namely LTCs with *c*onstant rate, *c*onstant distance, and *c*onstant locality.

In this work we construct such codes based on a new two-dimensional complex which we call a left-right Cayley complex. This is essentially a graph which, in addition to vertices and edges, also has squares. Our codes can be viewed as a two-dimensional version of (the one-dimensional) expander codes, where the codewords are functions on the squares rather than on the edges.



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