Correcting $k$ Deletions and Insertions in Racetrack Memory

One of the main challenges in developing racetrack memory systems is the limited precision in controlling the track shifts, that in turn affects the reliability of reading and writing the data. A current proposal for combating deletions in racetrack memories is to use redundant heads per-track resulting in multiple copies (potentially erroneous) and recovering the data by solving a specialized version of a sequence reconstruction problem. Using this approach, $k$-deletion correcting codes of length $n$, with $d \ge 2$ heads per-track, with redundancy $\log \log n + 4$ were constructed. However, the known approach requires that $k \le d$, namely, that the number of heads ($d$) is larger than or equal to the number of correctable deletions ($k$). Here we address the question: What is the best redundancy that can be achieved for a $k$-deletion code ($k$ is a constant) if the number of heads is fixed at $d$ (due to implementation constraints)? One of our key results is an answer to this question, namely, we construct codes that can correct $k$ deletions, for any $k$ beyond the known limit of $d$. The code has $4k \log \log n+o(\log \log n)$ redundancy for $k \le 2d-1$. In addition, when $k \ge 2d$, our codes have $2 \lfloor k/d\rfloor \log n+o(\log n)$ redundancy, that we prove it is order-wise optimal, specifically, we prove that the redundancy required for correcting $k$ deletions is at least $\lfloor k/d\rfloor \log n+o(\log n)$. The encoding/decoding complexity of our codes is $O(n\log^{2k}n)$. Finally, we ask a general question: What is the optimal redundancy for codes correcting a combination of at most $k$ deletions and insertions in a $d$-head racetrack memory? We prove that the redundancy sufficient to correct a combination of $k$ deletion and insertion errors is similar to the case of $k$ deletion errors.