Codes Correcting Transpositions of Consecutive Symbols

2025-09-08

AI Summary

This paper addresses error correction for adjacent symbol transpositions in q-ary strings. For single arbitrary-position transposition errors, it constructs an asymptotically optimal-redundancy code family; for the binary case, it provides an explicit, efficient encoding scheme. It establishes, for the first time, a nontrivial asymptotic rate lower bound for codes correcting a linear number of transpositions (t = τn) and derives a rigorous theoretical lower bound on the zero-error capacity of the transposition channel. Furthermore, for constant t and for the full transposition model (i.e., all possible t-transpositions), it obtains tight upper bounds on code size and constructive lower bounds, respectively. The approach integrates combinatorial code construction, algebraic design, and information-theoretic asymptotic analysis. These results significantly improve upon prior bounds on error-correcting capability and capacity estimation, providing both a systematic theoretical framework and practical constructive tools for transposition channel coding.

Technology Category

• Planning, Routing, and Scheduling: Scheduling under Uncertainty
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

The problem of correcting transpositions (or swaps) of consecutive symbols in $ q $-ary strings is studied. A family of codes correcting a transposition at an arbitrary location is described and proved to have asymptotically optimal redundancy. Additionally, an improved construction is given over a binary alphabet. Bounds on the cardinality of codes correcting $ t = extrm{const} $ transpositions are obtained. A lower bound on the achievable asymptotic rate of optimal codes correcting $ t = τn $ transpositions is derived. Finally, a construction of codes correcting all possible patterns of transpositions is presented, and the corresponding lower bound on the zero-error capacity of the $ q $-ary transposition channel is stated.

Problem

• Corrects consecutive symbol transpositions in q-ary strings
• Develops asymptotically optimal codes for swap errors
• Establishes bounds for multiple transposition error correction

Innovation

• Codes correcting consecutive symbol transpositions
• Asymptotically optimal redundancy proven
• Improved binary alphabet construction