In this paper, we present two reductions between variants of the Code Equivalence problem. We give polynomialtime Karp reductions from Permutation Code Equivalence (PCE) to both Linear Code Equivalence (LCE) and Signed Permutation Code Equivalence (SPCE). Along with a Karp reduction from SPCE to the Lattice Isomorphism Problem (LIP) shown by Bennett and Win (2024), our second result implies a reduction from PCE to LIP.

Mean-based reconstruction is a fundamental, natural approach to worst-case trace reconstruction over channels with synchronization errors. It is known that$\exp (\Theta (n^{1/3}))$traces are necessary and sufficient for mean-based worst-case trace reconstruction over the deletion channel, and this result was also extended to certain channels combining deletions and geometric insertions of uniformly random bits. In this work, we use a simple extension of the original complex-analytic approach to show that these results are examples of a much more general phenomenon. We introduceoblivious synchronization channels, which map each input bit to an arbitrarily distributed sequence of replications and insertions of random bits. This general class captures all previously considered synchronization channels. We show that for any oblivious synchronization channel whose output length follows a sub-exponential distribution either mean-based trace reconstruction is impossible or$\exp (O(n^{1/3}))$traces suffice for this task.

Mean-based reconstruction is a fundamental, natural approach to worst-case trace reconstruction over channels with synchronization errors. It is known that$\exp(O(n^{1/3}))$traces are necessary and sufficient for mean-based worst-case trace reconstruction over the deletion channel, and this result was also extended to certain channels combining deletions and geometric insertions of uniformly random bits. In this work, we use a simple extension of the original complex-analytic approach to show that these results are examples of a much more general phenomenon:$\exp(O(n^{1/3}))$traces suffice for mean-based worst-case trace reconstruction over any memoryless channel that maps each input bit to an arbitrarily distributed sequence of replications and insertions of random bits, provided the length of this sequence follows a sub-exponential distribution.