Dense locally testable codes cannot have constant rate and distance

TL;DR

This paper proves that dense locally testable codes cannot simultaneously have constant rate and distance, showing fundamental limitations on their structure and testing properties.

Contribution

It establishes that dense LTCs with certain query and density properties cannot be asymptotically good, advancing understanding of LTC limitations.

Findings

Dense 3-query LTCs with high-density testers cannot be ccc.

Higher-query LTCs with very high-density testers cannot be ccc.

Results hold under specific tester properties, like 'no weights' and 'last-one-fixed'.

Abstract

A q-query locally testable code (LTC) is an error correcting code that can be tested by a randomized algorithm that reads at most q symbols from the given word. An important question is whether there exist LTCs that have the ccc-property: constant relative rate, constant relative distance, and that can be tested with a constant number of queries. Such codes are sometimes referred to as "asymptotically good". We show that dense LTCs cannot be ccc. The density of a tester is roughly the average number of distinct local views in which a coordinate participates. An LTC is dense if it has a tester with density >> 1. More precisely, we show that a 3-query locally testable code with a tester of density >> 1 cannot be ccc. Moreover, we show that a q-query locally testable code (q>3) with a tester of density >> n^{q-2} cannot be ccc. Our results hold when the tester has the following two properties: 1) "no weights": Every q-tuple of queries occurs with the same probability. 2) "last-one-fixed": In every `test' of the tester, the value to any q-1 of the symbols determines the value of the last symbol. (Linear codes have constraints of this type). We also show that several natural ways to quantitatively improve our results would already resolve the general ccc-question, i.e. also for non-dense LTCs.