Perfect Codes Correcting a Single Burst of Limited-Magnitude Errors

Motivated by applications to DNA-storage, flash memory, and magnetic recording, we study perfect burst-correcting codes for the limited-magnitude error channel. These codes are lattices that tile the integer grid with the appropriate error ball. We construct two classes of such perfect codes correcting a single burst of length 2, where each error affects the corresponding position by increasing it by one, both for cyclic and non-cyclic bursts. We also present a generic construction that requires a primitive element in a finite field with specific properties. We then show that in various parameter regimes such primitive elements exist, and hence, infinitely many perfect burst-correcting codes exist.