LCC and LDC: Tailor-Made Distance Amplification and a Refined Separation

The Alon-Edmonds-Luby distance amplification procedure (FOCS 1995) is an algorithm that transforms a code with vanishing distance to a code with constant distance. AEL was invoked by Kopparty, Meir, Ron-Zewi, and Saraf (J. ACM 2017) for obtaining their state-of-the-art LDC, LCC and LTC. Cohen and Yankovitz (CCC 2021) devised a procedure that can amplify inverse-polynomial distances, exponentially extending the regime of distances that can be amplified by AEL. However, the improved procedure only works for LDC and assuming rate 1-1/(poly log n). In this work we devise a distance amplification procedure for LCC with inverse-polynomial distances even for vanishing rate 1/(poly log log n). For LDC, we obtain a more modest improvement and require rate 1-1/(poly log log n). Thus, the tables have turned and it is now LCC that can be better amplified. Our key idea for accomplishing this, deviating from prior work, is to tailor the distance amplification procedure to the code at hand. Our second result concerns the relation between linear LDC and LCC. We prove the existence of linear LDC that are not LCC, qualitatively extending a separation by Kaufman and Viderman (RANDOM 2010)