Generalized Repetition Codes and Their Application to HARQ

2025-11-19

AI Summary

Channel uncertainty under finite retransmission constraints causes decoding failures. Method: This paper proposes Generalized Retransmission Codes (GRCs), a novel coding framework that transcends conventional single-metric error-correcting codes by incorporating multiple metrics—e.g., Hamming distance and retransmission distance—and defining corresponding minimum distances. GRCs integrate hash-based feedback, CRC detection, and physical-layer FEC into a two-tier redundancy and adaptive feedback mechanism enabling multi-round collaborative error correction. Contribution/Results: Theoretically, we define and classify Type-I and Type-II optimal GRCs, deriving tight bounds for both. Constructively, we present efficient design algorithms to systematically generate optimal codes of both types. Experiments demonstrate that GRCs significantly improve the probability of correct decoding within a bounded number of retransmissions, establishing a new paradigm for ultra-low-latency, high-reliability communication.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Mixed Discrete/Continuous Search
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements

Abstract

The inherent uncertainty of communication channels implies that any coding scheme has a non-zero probability of failing to correct errors, making retransmission mechanisms essential. To ensure message reliability and integrity, a dual-layer redundancy framework is typically employed: error correction codes mitigate noise-induced impairments at the physical layer, while cyclic redundancy checks verify message integrity after decoding. Retransmission is initiated if verification fails. This operational model can be categorized into two types of repeated communication models: Type-I systems repeatedly transmit identical codewords, whereas Type-II systems transmit distinct coded representations of the same message. The core challenge lies in maximizing the probability of correct message decoding within a limited number of transmission rounds through verification-based feedback mechanisms. In this paper, we consider a scenario where the same error-correcting code is used for repeated transmissions, and we specifically propose two classes of generalized repetition codes (GRCs), corresponding to the two repeated communication models. In contrast to classical theory, we regard GRCs as error-correcting codes under multiple metrics--that is, GRCs possess multiple minimum distances. This design enables GRCs to perform multi-round error correction under different metrics, achieving stronger error-correction capabilities than classical error-correcting codes. However, the special structure of GRCs makes their construction more challenging, as it requires simultaneously optimizing multiple minimum distances. To address this, we separately investigate the bounds and constructions for Type-I and Type-II GRCs, and obtain numerous optimal Type-I and Type-II GRCs.

Problem

• Maximizing correct decoding probability in limited transmissions
• Designing generalized repetition codes with multiple minimum distances
• Optimizing error correction for Type-I and Type-II retransmission systems

Innovation

• Generalized repetition codes with multiple minimum distances
• Multi-round error correction under different metrics
• Optimal constructions for Type-I and Type-II systems