Order-Optimal Systematic Permutation Codes for Correcting t Deletions

2026-09-29

AI Summary

This study addresses the challenging problem of multiple-deletion error correction for permutation codes under both symbol-invariant and position-invariant deletion models. To tackle this, the authors construct systematic permutation codes that preserve the original sequence order through the insertion of redundant symbols. Furthermore, they propose an algebraic outer code framework based on integer moments and residual graph coloring to unify the treatment of both channel types. By leveraging redundancy markers to store inner correction syndromes alongside combinatorial techniques, the approach enables efficient encoding and decoding. The proposed scheme achieves near-optimal redundancy with polynomial time complexity. Additionally, this work establishes the equivalence of the two deletion channels under specific conditions, offering new theoretical insights into the design of robust permutation codes for deletion-prone environments.

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 $λ> t$.

Problem

• permutation codes
• multiple deletions
• symbol-invariant deletions
• permutation-invariant deletions
• systematic codes

Innovation

• permutation codes
• deletion correction
• systematic encoding
• integer moments
• residual graph coloring