Function-Correcting Lee-distance Codes for Symbol-Pair Read Channels

2026-10-07

AI Summary

This study addresses the optimization of error-correcting redundancy for q-ary symbol-pair read channels in high-density storage by extending the Lee metric to the Z_q space and establishing a theoretical framework for function-correcting codes. By integrating coding theory with combinatorics, this work derives the first classical code bounds for symbol-pair channels under the Lee metric, including the Plotkin and Gilbert–Varshamov bounds, and analyzes the relationship between irregular pair lengths and redundancy. Furthermore, explicit constructions are proposed for specific function classes. The research establishes tight bounds on optimal redundancy for linear and locally bounded functions, providing a rigorous theoretical foundation for enhancing the reliability of communication and storage systems.

Abstract

Function-correcting codes (FCCs) protect specified function evaluations of messages against errors while reducing the redundancy required for reliable communication. We study FCCs for symbol-pair read channels, motivated by high-density storage systems that read overlapping symbol pairs. Phase-shift keying (PSK) modulation is well-suited to such systems due to its bandwidth efficiency and noise robustness. While FCCs for symbol-pair read channels have been studied under the Hamming metric, the Lee metric is a more appropriate error model for $q$-ary PSK and, hence for $q$-ary symbol-pair read channels. We generalize the symbol-pair Lee distance, previously defined only over $\mathbb{Z}_4$, to $\mathbb{Z}_q$, $q\ge2$, and introduce function-correcting symbol-pair Lee-distance codes (FCSPLCs) over $\mathbb{Z}_q$, specializing to $q=2^m$, $m\ge1$, for $2^m$-ary PSK constellations. We investigate their redundancy requirements by introducing irregular-pair Lee-distance codes and relating the optimal redundancy of FCSPLCs to the shortest length of such codes. We derive Plotkin-type and Gilbert--Varshamov-type bounds on the optimal redundancy through lower and upper bounds on the shortest length of irregular-pair Lee-distance codes. For bijective functions, we obtain corresponding Plotkin-type and Gilbert--Varshamov-type bounds for classical Lee metric codes for symbol-pair read channels over $\mathbb{Z}_q$, which, to the best of our knowledge, are the first such bounds for the Lee metric symbol-pair setting. We then specialize the FCSPLC framework to pair-locally bounded functions, the Pair-Lee weight function, and the Pair-Lee weight distribution function, giving explicit constructions and corresponding bounds on the optimal redundancy. Finally, for linear functions, we derive a Plotkin-type lower bound on the optimal redundancy.

Problem

• Function-correcting codes
• Symbol-pair read channels
• Lee distance
• Phase-shift keying
• Redundancy bounds

Innovation

• Function-correcting codes
• Symbol-pair read channels
• Lee distance
• Irregular-pair Lee-distance codes
• Redundancy bounds