Deletion-Correcting Codes for the $\ell$-Symbol Read Channel

This paper studies deletion-correcting codes for the $\ell$-symbol read channel, whose noiseless output is the vector of all consecutive $\ell$-mers of a transmitted sequence. This model is motivated by overlapping-read mechanisms arising in nanopore sequencing, racetrack memories with consecutive read heads, and related sequence-labeling problems. We consider an adversarial setting in which a fixed number of $\ell$-mers are deleted from the read vector. Our first contribution is a structural characterization of the effect of such deletions: after a minimum number of $\ell$-mers are inserted to restore consistency, the resulting sequence is obtained from the transmitted sequence by deleting symbols from certain periodic substrings; when $t\le \ell-2$, these deletions correspond to complete minimum periods. Based on this characterization, we introduce check patterns and construct $\ell$-read deletion-correcting codes via power-sum syndromes. For every $\ell\ge2$, we obtain single-deletion correcting codes with redundancy $\log\lfloor (n+2\ell)/(\ell-1)\rfloor$. For $2\le t\le \ell/2$, we construct $q$-ary $\ell$-read $t$-deletion correcting codes with redundancy $t\log n+O(1)$, and for $\ell=2t-1$ with $t\ge3$, we construct codes with redundancy $(2t-1)\log n+O(1)$. We also study the sporadic parameter pairs $(\ell,t)\in\{(2,2),(3,2),(3,3)\}$ and obtain improved constructions, including binary $\ell$-read $2$-deletion correcting codes with redundancy $2\log n+O(1)$ for $\ell=2,3$, a non-binary $3$-read $2$-deletion correcting code with redundancy $3\log n+O(1)$, a binary $3$-read $3$-deletion correcting code with redundancy $5\log n+O(1)$, and a non-binary $3$-read $3$-deletion correcting code with redundancy $7\log n+O(1)$.