Generalized Repetition Codes and Their Application to HARQ

TL;DR

The paper tackles reliability and latency challenges in HARQ by introducing generalized repetition codes (GRCs) that enable multi-round error correction under multiple metrics. It develops Type-I and Type-II GRCs, each leveraging different generator structures to realize multi-round decoding through sub-block metrics, notably the sub-block distance hierarchy (SBDH) and sub-Hamming distance hierarchy (SHDH). The authors derive a Griesmer-type bound for Type-I GRCs, establish SBDH/SHDH-based construction methods from cyclic, extended cyclic, and quasi-cyclic codes, and connect Type-II GRCs to general linear codes for broader design flexibility. They demonstrate that GRCs can outperform classical repetition, b-symbol codes, and IR-linear codes in frame error rate under HARQ scenarios, offering a versatile framework for practical, multi-round HARQ code design and decoding strategies. The work provides both theoretical bounds and concrete constructions, underscoring the potential of Type-II GRCs to exceed traditional limits via multi-metric block coding and structured code families.

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.

Figures (5)

• Figure 1: Two types of repeated communication channels: Alice repeatedly transmits the same encoded version $\mathbf{c}$ (resp. different encoded version $\mathbf{c}_i$) of message $\mathbf{u}$ to Bob through a noisy channel. Bob decodes the received codewords $\mathbf{y}_i$ to generate a list of candidate codewords $L_I$ (resp. $L_{II}$). If $L_I$ or $L_{II}$ contains the correct message $\mathbf{u}$, then decoding succeeds.
• Figure 2: FER curves of binary cyclic Golay codes under $b$-symbol metric with different $b$: The signal-to-noise ratio (SNR) quantifies the ratio of signal power to noise power, expressed in decibels (dB) as $10 \log_{10}(P_{\text{signal}} / P_{\text{noise}})$.
• Figure 3: Modified Type-I repeated communication model using Type-I GRCs: Once Bob receives $m\ge 2$ codewords: $\mathbf{y}_1,\ldots,\mathbf{y}_{m}$, Bob takes Chase Combining and permutations to them get $\hat{\mathbf{y}}$ and $\mathbf{y}_i^{\prime}=\mathbf{y}_iA_{i-1}$, respectively, where $i\ge 2$ and $A_i\in Ps_{n}(\mathbb{F}_q)$.
• Figure 4: Modified Type-II repeated communication model using Type-II GRCs: Alice performs a non-singular linear transformation $B_i$ on message $\mathbf{u}$ before retransmitting it each time. i.e., $\mathbf{c}_i=\mathbf{u}B_{i-1}G$ for $i\ge 2$, where $B_i \in GL_k(\mathbb{F}_q)$.
• Figure 5: Comparison of FER between GRCs and Classical Repetition Codes under ML Decoding Algorithm under SNR=$-5$ dB. (a) Type-I GRC $\mathcal{C}_I$ with different decoding depths VS. classical repetition code $C_1$ and $b$-symbol code $C_2$. (b) Type-II GRC $\mathcal{C}_{II}$ VS. IR-linear code $C_3$.

Theorems & Definitions (26)

• Lemma 1
• Lemma 2
• Lemma 3
• Definition 1
• Definition 2
• Example 1
• Definition 3
• Lemma 4
• Theorem 1
• Example 2