Error-Correcting Weakly Constrained Codes: Constructions and Achievable Rates

2026-05-09

AI Summary

This work proposes a novel weakly constrained coding framework that, for the first time, integrates frequency-constrained pattern allowance with error-correction capability. Traditional constrained codes merely forbid specific patterns and lack control over the occurrence frequencies of permitted patterns, while also struggling to maintain robust error-correction performance. In contrast, the proposed approach constructs capacity-achieving weakly constrained codebooks via Eulerian cycle-based methods and employs a puncturing technique to obtain codes with linear minimum distance and positive code rate. Furthermore, a concatenated architecture is designed to enable efficient encoding and decoding with polynomial-time complexity. The resulting scheme not only guarantees strong theoretical performance but also offers a practical pathway for implementation.

Abstract

We investigate weakly constrained codes, in which specific patterns occur with prescribed frequencies rather than being strictly forbidden as in conventional constrained coding. We propose a capacity-achieving construction of a weakly constrained codebook based on Eulerian cycles. We then obtain, via expurgation, weakly constrained codes with linear minimum distance and positive rate, and analyze the rates achievable. Finally, we propose a practical concatenated code construction that supports polynomial-time encoding and decoding.

Problem

• weakly constrained codes
• error-correcting
• achievable rates
• minimum distance
• constrained coding

Innovation

• weakly constrained codes
• Eulerian cycles
• expurgation
• concatenated codes
• achievable rates