Polar IR-HARQ Scheme

• Polar IR-HARQ scheme is a set of hybrid automatic repeat request methods that use polar codes to deliver incremental redundancy, not mere repetition.
• It employs mechanisms like puncturing, dynamic freezing, and copy-bit extension to maintain nested reliability and achieve rate compatibility across transmissions.
• The approach supports various channel models and decoding architectures, achieving near-capacity performance and flexible modulation in practical systems.

Polar IR-HARQ denotes a family of hybrid automatic repeat request schemes in which polar-coded retransmissions provide incremental redundancy rather than mere repetition, so that decoding uses cumulative observations, cumulative mutual information, or an explicitly extended polar structure. The central design problem is rate compatibility under polarization: the retransmission mechanism must preserve, reshape, or re-exploit the polarized reliability ordering while allowing the effective rate to decrease across rounds. The literature addresses this problem through universal super polar codes for parallel and block-fading channels, puncturing- and repetition-based rate matching, quasi-uniform puncturing with dynamically frozen bits, extension-based copy-bit relocation, inter-transmission masking, segmented CRC-aided decoding, and rateless capacity-aware scheduling (Li et al., 2022, El-Khamy et al., 2017, Yuan et al., 2018).

1. Information-theoretic basis

In the canonical IR-HARQ viewpoint, the first transmission sends a higher-rate polar codeword, and each subsequent transmission contributes additional useful information. For block fading, a particularly clean formulation models the $i$-th HARQ round as a channel block of capacity $r_i$, with successful decoding after $M$ rounds whenever

$r_1 + r_2 + \cdots + r_M \ge R,$

where $R$ is the information rate of the first transmission; in the large-blocklength regime, the corresponding polar construction is capacity-achieving when decoding is performed jointly across all received pieces (Li et al., 2022). This is the information-theoretic meaning of polar IR-HARQ: retransmissions are not repetitions of a fixed codeword in the Chase-combining sense, but new contributions to cumulative mutual information.

A finite-length engineering formulation appears in early rate-compatible polar work through the $(N,K,M)$ rate-compatible polar code, where $K$ is the number of information bits, $M$ is the number of polar bits produced by a punctured polar encoder, and $N-M$ are repetition bits. In that setting, the receiver decodes after each attempt using all received bits so far, and throughput efficiency is defined as

$\eta = \frac{E[K]}{E[N]}.$

This formulation makes explicit the dual objectives that have remained central in later work: maintaining a useful nested family of codes across retransmissions and maximizing throughput rather than only minimizing single-shot error rate (Chen et al., 2013).

A recurrent misconception is to identify HARQ for polar codes with Chase combining. The literature consistently separates the two. Chase combining retransmits the same coded bits and adds log-likelihood ratios, whereas IR-HARQ changes the effective code by sending new parity bits, newly exposed positions of a mother code, or new constrained copies of weak information bits. The receiver therefore benefits not only from accumulated energy but also from a lower effective code rate or a stronger global code constraint.

2. Coding mechanisms for rate compatibility

All major polar IR-HARQ constructions inherit the standard polar transform

$r_i$0

but they differ in how they maintain a nested reliability structure across retransmissions. A central mechanism is nested information sets on degraded or progressively extended channels. In the universal parallel-channel construction, for a sequence of degraded BMS channels $r_i$1, one constructs nested information sets

$r_i$2

which allows any partial collection of transmissions with sufficient total capacity to support decoding (Li et al., 2022).

A practical line of work achieves rate compatibility by puncturing and repetition. The rate-compatible wireless-channel design based on progressive puncturing search runs puncturing-order optimization on a short base code, lifts that order through two-step polarization, permutes the encoded bits according to the reverse puncturing order, and then reads them column-wise from a rate-matching matrix. This produces a single mother code that supports arbitrary effective lengths by puncturing when fewer than $r_i$3 bits are transmitted and repetition when more than $r_i$4 bits are required, while preserving the compound polarization structure across longer codes (El-Khamy et al., 2015).

A second practical mechanism is quasi-uniform puncturing. For a desired $r_i$5 QUP-polar code, the first $r_i$6 coded bits are punctured in bit-reversal order, their mutual information is initialized to zero during Gaussian-approximation construction, and the corresponding punctured positions are treated as zero-LLR inputs at the decoder. In the flexible IR-HARQ construction based on QUP, dynamic freezing is then used to tie different effective lengths together by constraints of the form

$r_i$7

so that a later lower-rate code remains consistent with earlier transmissions while allowing arbitrary per-round lengths rather than only powers of two (Yuan et al., 2018).

A third mechanism is extension by copy-bit relocation. In the extension scheme, a failed $r_i$8-length transmission is upgraded to a $r_i$9-length polar code, and weak information bits from the original block are copied into more reliable positions of the extended part. In the final $M$0-length decoding graph, the newer occurrence is treated as the information-bearing bit, while the older occurrence is treated as frozen to the same value; this allows the combined transmissions to behave nearly like a directly designed longer polar code (Ma et al., 2017).

3. Principal construction paradigms

The term “polar IR-HARQ scheme” covers several distinct construction philosophies rather than a single canonical design.

Paradigm	Representative mechanism	Distinctive property
Universal super polar code (Li et al., 2022)	Parallel-channel decomposition with coordinated sub-channel polarization	Decoding threshold depends on sum capacity
Circular-buffer rate matching (El-Khamy et al., 2017)	Reverse puncturing order, column-wise readout, shifted start columns	Arbitrary rate by puncturing or repetition
RCP family (Chen et al., 2013)	Punctured polar code plus selective repetition of weak information bits	Fixed polar core, variable redundancy
Flexible QUP with dynamic freezing (Yuan et al., 2018)	Arbitrary-length puncturing plus dynamically frozen constraints	No per-round power-of-two restriction
Copy-bit extension (Ma et al., 2017)	Weak bits copied to stronger positions in the extended block	Combined rounds form a longer polar code
ARUM (Chen et al., 2018)	XOR masks across transmissions with active-bit relocation	Incremental channel polarization
Capacity-aware rateless scheduling (Zhang et al., 29 May 2026)	Nested parity-check construction and reverse bit-mapping	Continuous $M$1

These paradigms answer the same question in different ways. The universal parallel-channel approach treats HARQ rounds as virtual parallel channels and proves a sum-capacity result. Circular-buffer schemes start from a single mother code and expose different segments or column offsets across rounds. QUP-based designs generalize puncturing to arbitrary lengths and use dynamic constraints to preserve cross-round consistency. Extension-based schemes enlarge the polar transform itself. ARUM adds a second polarization dimension across transmissions through binary masks. Rateless schedule-based designs preserve a single mother code but move decoding authority to a capacity-aware schedule rather than a fixed left-to-right order.

A second common misconception is that polar IR-HARQ must be puncturing-based. The literature shows otherwise. Some schemes are puncturing-centered, some are extension-centered, some operate through repeated copies of selected information bits, and some rely on cross-round parity constraints or decoding-order adaptation. The unifying feature is not the specific rate-matching primitive but the preservation of a reliable nested decoding relation across HARQ rounds.

4. Decoding architectures and receiver operation

Receiver design is as important as encoder design in polar IR-HARQ. The standard options remain successive cancellation and successive cancellation list decoding, often augmented by CRC. In circular-buffer and QUP-based systems, after each HARQ round the receiver deinterleaves the new observations, places them into the mother-code positions, sets punctured positions to zero-LLR inputs, sums LLRs for repeated bits, and runs SC or SCL on the cumulative observation vector. For block-fading and universal parallel-channel designs, the receiver instead interprets the accumulated rounds as outputs of parallel channels and performs joint polar decoding across all received pieces (Li et al., 2022, El-Khamy et al., 2017).

Segmented decoding introduces a different granularity. In tailored-CRC-aided SCL, the codeword is partitioned into segments, each segment receives a CRC length derived from its virtual length, and decoding proceeds segment by segment. The HARQ extension of that decoder retransmits only a failed segment; the new observation is merged with the old one by maximum ratio combining, and the segment is re-decoded. This is a segment-wise IR style: redundancy is requested and consumed at segment granularity rather than whole-codeword granularity (Zhou et al., 2018).

ARUM changes both encoding and decoding order. Each transmission is an independent polar codeword embedded into a common alignment length and XOR-masked with prior codewords according to an upper-triangular binary matrix. The masking induces an inter-transmission polarization transform, after which less reliable active bits in earlier transmissions are relocated to more reliable positions in the latest transmission. Decoding is therefore performed from the latest block to the earliest, and relocated bits in older blocks become non-active bits: they are treated analogously to frozen bits, but with values determined by the relocated copies rather than fixed zeros (Chen et al., 2018).

Fast decoder support has recently become a distinct subproblem. Matrix-extension IR-HARQ introduces parity-check frozen bits whose values are constrained by other information bits, which breaks the assumptions behind conventional special-node SC decoding. A modified SC architecture restores Rate-0, REP, REP-2, SPC, SPC-2, RPC, and PCR node handling under these parity-check constraints. For a polar code of length 2048, the modified scheme reduces node traversals by 72% relative to an unmodified SC IR-HARQ decoder, and the reported simulations show no degradation in FER performance (Jalaleddine et al., 4 Dec 2025).

5. Channel models, modulation, and system embeddings

Polar IR-HARQ has been developed not only for binary-input AWGN, but also for block fading, parallel Gaussian channels with non-binary inputs, BICM with higher-order modulation, multilevel polar-coded modulation, and MIMO. In the universal construction, $M$2 parallel Gaussian channels are decomposed into binary sub-channels, and a super polar code is designed across all of them. The HARQ interpretation then treats the first $M$3 HARQ rounds as $M$4 active parallel channels and the remaining $M$5 rounds as channels of capacity zero. This yields a capacity-achieving block-fading HARQ scheme and, in the MIMO setting, a design that requires only sum-rate feedback rather than per-layer rate feedback (Li et al., 2022).

For BICM and higher-order modulation, circular-buffer rate-matched polar codes integrate a column interleaver and a bit-mapping strategy so that each column is assigned, as far as possible, to a single BICM bit-level. This preserves a compound polarization structure under QPSK, 16-QAM, and 64-QAM. In IR mode, the starting column is shifted across rounds, and the column-to-bit-level assignment is shifted in parallel, which provides additional mapping diversity beyond the appearance of new code bits (El-Khamy et al., 2017).

Multilevel polar-coded modulation extends the same logic to ASK and QAM through level-wise polar codes. The flexible QUP-based IR-HARQ scheme applies to biAWGN and to higher-order modulation via multilevel polar coding with set-partition labeling, using MI-DGA to allocate reliability across bit-levels. A related throughput-based design for multilevel polar coded modulation develops set-partitioning-based bit-to-symbol mapping and average-LLR-based GA construction for NC and CC HARQ, and explicitly states that the same throughput-maximizing design principles are applicable to IR, although the main detailed IR development there is for BPSK (Yuan et al., 2018, Khoshnevis et al., 2018).

This variety of channel models matters because the “new redundancy” of IR-HARQ can mean different objects in different embeddings: additional time blocks in block fading, additional columns in a circular buffer, additional ASK/QAM symbols in polar-coded modulation, or additional MIMO-layer allocations under sum-rate-aware scheduling.

6. Performance, trade-offs, and unresolved issues

The performance record of polar IR-HARQ is heterogeneous because the schemes solve different problems. In the early rate-compatible polar family based on puncturing plus selective repetition, the HARQ scheme over BAWGNC with $M$6 is reported to operate within about $M$7–$M$8 dB of channel capacity under low-complexity SC decoding, even though standalone RCP codes are reported as $M$9–$r_1 + r_2 + \cdots + r_M \ge R,$0 dB worse than WCDMA turbo codes around BLER $r_1 + r_2 + \cdots + r_M \ge R,$1; the gain comes from the HARQ protocol rather than from the single-shot code alone (Chen et al., 2013).

For circular-buffer rate matching, the representative 16-QAM AWGN example with $r_1 + r_2 + \cdots + r_M \ge R,$2, $r_1 + r_2 + \cdots + r_M \ge R,$3, and SC decoding reports about $r_1 + r_2 + \cdots + r_M \ge R,$4 dB gain for IR-HARQ over Chase combining at a given target error rate. The same framework also supports adaptive modulation and coding under fast fading and competitive comparisons against LTE turbo and SC-QC LDPC baselines (El-Khamy et al., 2017). The flexible QUP and dynamically frozen construction reports performance close to directly designed QUP-polar codes across multiple cumulative lengths and, in an 8-ASK polar-coded-modulation example, about $r_1 + r_2 + \cdots + r_M \ge R,$5 dB gain over 5G-LDPC after the second to fourth transmissions at the same FER (Yuan et al., 2018).

Decoder-aware and architecture-aware refinements also produce measurable gains. Tailored-CRC segmented HARQ adds about $r_1 + r_2 + \cdots + r_M \ge R,$6 dB at $r_1 + r_2 + \cdots + r_M \ge R,$7 over non-HARQ TCA-SCL in the reported experiments, while ARUM is reported to achieve near-optimal coding gain; after four transmissions it is slightly worse than the directly constructed $r_1 + r_2 + \cdots + r_M \ge R,$8 polar code by less than $r_1 + r_2 + \cdots + r_M \ge R,$9 dB, and the gain of the full Arıkan mask over the lower-complexity FL mask is described as trivial (Zhou et al., 2018, Chen et al., 2018). The recent rateless capacity-aware framework reports that its codes match the coding gain of independently optimized fixed-rate codes across the entire range of rates and lengths, with validated hardware overheads of about 23% in area and about 22% in power relative to a conventional polar SCL decoder in the cited implementation (Zhang et al., 29 May 2026).

A recurring technical tension runs through the literature. Asymptotic formulations can prove sum-capacity achievement or vanishing error probability under large $R$0 and ideal feedback, but practical designs must confront finite-length penalties, rate-matching granularity, CRC placement, list size, and hardware throughput. Another recurring tension is flexibility versus structural purity: some schemes preserve the standard polar graph but impose power-of-two extensions, whereas others allow arbitrary lengths or arbitrary $R$1 at the price of dynamic freezing, parity-copy constraints, or nonstandard decoding schedules. This suggests that “polar IR-HARQ scheme” is best understood as a research area organized around a shared rate-compatibility objective, rather than as a single fixed encoder-decoder recipe.