Correcting Bursty/Localized Deletions: A New Error-Position-Estimation Code

Abstract: Codes correcting bursts of deletions and localized deletions have garnered significant research interest in recent years. One of the primary objectives is to construct codes with minimal redundancy. Currently, the best known constructions of $q$-ary codes correcting a burst of at most $t$ deletions ($(\le t)$-burst-deletion correcting codes) achieve redundancy $\log n+8\log\log n+o(\log\log n)$ (for any $q$ and $t$) or $\log n+t\log\log n+O(1)$ (for even $q$). For codes correcting single $t$-localized-deletion ($t$-localized-deletion correcting codes), state-of-the-art constructions attain redundancy $\log n+O\parenv{t(\log\log n)^2}$ (for any $q$ and $t$) or $\log n+2t\log\log n+O(1)$ (for even $q$). Here, $n$ denotes the code-length, and $q$ and $t$ are fixed. These codes employ a position-estimation component to approximate error positions, augmented by additional constraints that enable error-correction given the information about error positions. In this work, we select codewords from the set of sequences whose differential sequences are strong-$(\ell,\epsilon)$-locally-balanced. By imposing a VT-type constraint and an $L_1$-weight constraint on the differential sequences of codewords, we construct novel position-estimation codes. When $q\ge 2$ and $t<q$, or $q$ is even and $t<2q$, this approach gives a $q$-ary $(\le t)$-burst-deletion correcting code and a $t$-localized-deletion correcting code with redundancy $\log n+(t-1)\log\log n+O(1)$. In addition to improving previous redundancy, the method is new and our position-estimation codes are simpler than those in previous works. Finally, we give an efficient encoder to encode an arbitrary input sequence into a sequence whose differential sequence is strong-$(\ell,\epsilon)$-locally-balanced. To our knowledge, no prior algorithm for this specific task has been reported.