I think the abstract of this paper spewaks for itself and the introduction is also a pleasure to read. Still, let me make a few comments.
First, the impossibility result of BGS03 rules out not only good codes (i.e., linear distance and constant rate) but rather any code that is not extremely bad. Hence, the dichomomy established in the current paper is between parameters that allow a good code and ones that allow only extremely bad codes.
Second, as stated clearly in the abstract and introduction, the new results build on a prior work of First and Kaufman, which I neglected to recommend at the time.
Ben-Sasson, Goldreich and Sudan showed that a binary error correcting code admitting a 2-query tester cannot be good, i.e., it cannot have both linear distance and positive rate. The same holds when the alphabet is a finite field F, the code is F-linear, and the 2-query tester is -linear. We show that those are essentially the only limitations on the existence of good locally testable codes (LTCs). That is, there are good 2-query LTCs on any alphabet with more than 2 letters, and good 3-query LTCs with a binary alphabet. Similarly, there are good 3-query F-linear LTCs, and for every F-vector space V of dimension greater than 1, there are good 2-query LTCs with alphabet V whose tester is F-linear. This completely solves, for every q and alphabet Sigma (resp. F-vector space), the question of whether there is a good q-query LTC (resp. F-LTC) with alphabet Sigma. Our proof builds on the recent good 2-query F-LTCs of the first author and Kaufman, by establishing a general method for reducing the alphabet size of a low-query LTC.
See arXiv 2512.16082.