communication protocol engineers

Abstract: We consider the following data compression problem. Given a string $x \in \{0,1\}^m$ of Hamming weight at most $n$, compress it into a shorter string $y \in \{0,1\}^s$ so that any bit $x_i$ of $x$ can be retrieved without any error using at most $t$ quantum queries to the standard oracle encoding of $y$. If queries are allowed to be adaptive we show how optimal compression up to a logarithmic factor can be achieved. If the queries are required to be made non-adaptively, we show schemes whose space is optimal in its dependence on $m$ except for a logarithmic factor, and is at most quadratically worse when compared to the optimum in its dependence on $n$.

Abstract: This paper investigates the construction of full-systematic permutation codes capable of correcting multiple deletions under two complementary models, namely symbol-invariant deletions (SIDs), where surviving symbol values are preserved, and permutation-invariant deletions (PIDs), where the surviving sequence is standardized to a permutation. For any fixed integer $t \ge 1$ and all sufficiently large message lengths $n$, our proposed encoders map any message permutation of length $n$ to a codeword by inserting distinct redundancy symbols while strictly preserving the sequence order of the original message symbols. The proposed constructions correct up to $t$ deletions using $7t-1$ redundancy markers for PIDs and $4t$ redundancy markers for SIDs, achieving redundancies of $(7t-1)\log n + O_t(1)$ bits and $4t\log n + O_t(1)$ bits, respectively. Both code families are uniformly constructible, encodable, and decodable in $n^{O(t)}$ time. The underlying framework stores an inner deletion-correcting syndrome in the relative positions of redundancy markers via an algebraic outer code based on integer moments and residual graph coloring. We further extend this framework to fixed-composition and strictly $λ$-regular multipermutations, proving that the PID and SID channels coincide whenever the common multiplicity satisfies $λ> t$.