Linial’s Algorithm and Systematic Deletion-Correcting Codes

In this paper, we present a universal method to construct systematic codes correcting a constant number of errors that are traditionally hard to handle, such as insertions and deletions. Our method is based on Linial’s distributed graph coloring algorithm, and the codes have polynomial-time encoding and decoding complexity, with a redundancy that is about twice the Gilbert-Varshamov bound. As an application, for q = O(poly(n)), where q is the size of the alphabet and n is the code length, we construct systematic codes correcting t deletions and s substitutions with redundancy (4t + 4s) log n + (3t + 6s) log q + O(log log n) and systematic codes correcting t deletions or s substitutions with redundancy 4 max{t, s} log n+3 max{t, 2s} log q+O(log log n). We also show that the celebrated ‘syndrome compression’ technique, proposed by Sima et al. (ISIT 2020), can be viewed as an application of Linial’s algorithm. Thus our method is a generalization of syndrome compression.