Locally Testable Codes with Constant Rate, Distance, and Locality
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.
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LTCs were initially studied as important components of probabilistically checkable proofs (PCP), 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.
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An outstanding open question has been whether there exist 鈥c3-LTCs鈥, namely LTCs with constant rate, constant distance, and constant locality.
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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.