DETERMINISTIC CONSTRUCTION OF SYNCHRONIZATION STRING OVER SMALL ALPHABET

Synchronization string, first introduced by Haeupler and Shahrasbi [10], is a strong tool in construction of error correcting codes for insertion and deletion errors (insdel codes). Synchronization string provides a way to encode the indices of symbols in a string and makes it possible to transfer synchronization errors to easier half errors, which is much better understood. In this paper, we improve the construction in [10] in the following aspects: • We achieve a smaller alphabet size, reduce the alphabet size from O(ε−4) in [10] to O(ε−2). • We give an efficient deterministic construction of synchronization string over alphabet of size O(ε−3). • We give a near linear deterministic construction of synchronization string over alphabet of size O(ε−4). This algorithm runs in O(n log log n). Independently, Haeupler and Shahrasbi give a linear, deterministic construction for synchronization string over alphabet of ε−O(1) in their work [11]. • We introduce a combinatorial object called synchronization circle which enhances the property of synchronization string.