Beyond 1$\to$N Decoding: Capacity-Aware Rateless Polar Codes for IR-HARQ

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Beyond 1$\to$N Decoding: Capacity-Aware Rateless Polar Codes for IR-HARQ

Unknown · 2026 · arxiv_cs

arXiv CS · Papers · License: Open Access · 2026

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distributedsystems protocols

networking, internet, protocols, distributed systems

ZHANG, WANG AND TONG: BEYOND 1→N DECODING: CAPACITY-AWARE RATELESS POLAR CODES FOR IR-HARQ

Beyond 1→N Decoding: Capacity-Aware Rateless Polar Codes for IR-HARQ

arXiv:2605.30885v1 [cs.IT] 29 May 2026

Huazi Zhang, Xianbin Wang, Jiajie Tong, Jun Wang, Wen Tong,

Abstract—This paper introduces a novel framework for polar codes, designed for ﬂexible Incremental Redundancy Hybrid Automatic Repeat Request (IR-HARQ). By generalizing the decoding order beyond the standard 1 → 𝑁 sequence, we enable a capacity-aware scheduling strategy that prioritizes the decoding of reliable subblocks. The framework integrates nested parity-check polar construction and reverse bit-mapping to support continuous and arbitrary transmission lengths 𝐸 ∈ [𝑁min , 𝑁max ]. Simulation results show that the proposed rateless codes match the coding gain of independently optimized ﬁxedrate codes across the entire range of rates and lengths. With a validated hardware implementation, this work provides a practical solution for next-generation wireless data channels. Index Terms—Beyond 6G, extreme connectivity, channel coding, KPI, tradeoff.

I. I NTRODUCTION A. Motivation Hybrid automatic repeat request (HARQ) combines forward error correction with retransmissions to improve spectral efﬁciency and reliability on wireless data channels. Incremental redundancy HARQ (IR-HARQ) transmits new parity bits in each retransmission instead of simply repeating the same codeword, which signiﬁcantly increases spectral efﬁciency and robustness [1]. IR-HARQ integrates tightly with link adaptation and enables wireless systems to operate efﬁciently over a wide range of SNRs and mobility conditions. B. Rateless IR-HARQ The rateless IR-HARQ property is essential to modern wireless communication systems. It ensures that transmission rates are always aligned with the instantaneous channel capacity without requiring complex, ﬁxed-rate code constructions. Consider a code block of 𝐾 information bits to be encoded once into a mother codeword of length 𝑁max bits All authors are with Huawei Technologies Co. Ltd. (Email: {zhanghuazi, wangxianbin1, tongjiajie, justin.wangjun, tongwen}@huawei.com).

Encoding

Info bits Code bits t 6th tx P

5th tx 4th tx

Circular buffer

3rd tx

1st tx 2nd ttx x

1st Rateless transmissions with multiple redundancy versions

2nd 3rd 4th 5th 6th

…

Fig. 1. Rateless IR-HARQ

(the maximum amount of coded redundancy available for that block). The 𝑁max coded bits are written sequentially into a circular buffer. At each (re)transmission, the transmitter does not reencode. Instead, it reads out 𝐸 𝑡 bits from the circular buffer (where 𝐸 𝑡 can vary arbitrarily from transmission to transmission, depending on the scheduled resources at time 𝑡 ) and sends them over the channel. When the end of the buffer is reached, reading continues from the beginning (wrap around). The total number of coded bits transmitted after 𝑇 transmissions is 𝑇 Õ 𝐸 (𝑇 ) = 𝐸𝑡 . 𝑡=1

The effective code rate after combining all received redundancy up to 𝑇 transmissions is then 𝐾 𝑅 (𝑇 ) = (𝑇 ) . 𝐸 Because 𝐸 𝑡 and the number of transmissions 𝑇 are not ﬁxed in advance, the scheme is rateless: the effective code rate is not predetermined at encoding [2]. The rateless IR-HARQ scheme, as illustrated in Fig. 1 is required to achieve 1) Fine-granularity, on-demand rate adaptation (ﬂexibility): support ﬁne-granularity incremental redundancy with arbitrary 𝐸 𝑡 and arbitrary 𝑇 , allowing an essentially continuous set of effective code rates 𝑅 (𝑇 ) = 𝐾/𝐸 (𝑇 ) (up to the limit 𝐸 (𝑇 ) ≤ 𝑁max );

ZHANG, WANG AND TONG: BEYOND 1→N DECODING: CAPACITY-AWARE RATELESS POLAR CODES FOR IR-HARQ

Systematic bit

1 1st parity

Puncture for rate>1/3 ʌ

1 2nd parity D

Repetition for rate<1/3

Fig. 2. Turbo-based IR-HARQ (LTE)

2) Rate-compatibility / nested codewords: any lowerrate codeword is an extension of any higher-rate one. The family of codes {C (𝐾, 𝐸) : 𝐾 ≤ 𝐸 ≤ 𝑁max } must be nested (rate-compatible) in the sense that if 𝐸 (1) < 𝐸 (2) , then the length-𝐸 (1) codeword is the preﬁx (or a ﬁxed-index subset) of the length-𝐸 (2) codeword; 3) Near-optimal performance at every effective rate (no loss from nesting): the performance at rate 𝑅 (𝑇 ) is not signiﬁcantly worse than that of an independently designed, non-rateless code of the same rate and blocklength. II. E XISTING IR-HARQ SCHEMES IN 3GPP STANDARDS

In this section, we brieﬂy review existing IR-HARQ schemes deployed from 3G to 5G, and discuss whether they have fully realized the rateless IR-HARQ capability. A. 3G/4G Turbo codes IR-HARQ was ﬁrst widely deployed in 3G systems [3], where turbo codes [4] with rate-compatible puncturing support adaptive transmission on data channels by starting from a relatively high-rate turbo code and progressively sending additional parity bits upon NACKs (negative-acknowledgments). This mechanism was reused in LTE (4G) [5], still based on turbo codes but employing a uniﬁed circular-buffer rate-matching structure that deﬁned four redundancy versions, i.e., 𝑅𝑉 = 0, 1, 2, 3. As shown in Fig. 2, the LTE turbo code is based on two identical recursive systematic convolutional (RSC) component encoders concatenated in parallel through an interleaver. Each information bit is transmitted systematically and is protected by one parity bit from each component encoder. As a result, the “mother” code naturally has a nominal code rate of 1/3. LTE turbo codes, including the code distance and decoding threshold, are

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濃濃濃濃濃濃濃濃濃濃濃濃濃濃濃濃濃濃濃濃濃濃濃濃濃濃濃濃濃濃濃濃濃濃濃濃濃濃濃濄濃濃

濃濃濃濃濃濃濃濃濃濃濃濃濃濃濃濃濃濃濃濃濃濃濃濃濃濃濃濃濃濃濃濃濃濃濃濃濃濃濃濃濄濃

濃濃濃濃濃濃濃濃濃濃濃濃濃濃濃濃濃濃濃濃濃濃濃濃濃濃濃濃濃濃濃濃濃濃濃濃濃濃濃濃濃濄

Can be further extended …

Fig. 3. LDPC-based IR-HARQ (5G Base Graph 2)

optimized to achieve near-optimal performance around this mother code of 1/3. To support higher code rates in IR-HARQ, LTE employs extensive puncturing of the parity streams to effectively increase the rate above 1/3. However, at these high rates the resulting punctured turbo codes deviate substantially from the properties of the underlying mother code. Consequently, turbo codes exhibit signiﬁcant performance loss for high code rates, e.g., error ﬂoors due to low-weight codewords. On the other hand, for code rates below 1/3, extra redundancy is offered by simple repetition rather than by generating new, independent parity symbols. It only provides additional energy gain. Since turbo codes do not provide near-optimal performance at every effective rate, turbo-based IR-HARQ cannot fully exploit the potential of rateless IR-HARQ. B. 5G LDPC codes Starting with 5G NR [6], LDPC codes [7] are adopted for data channels. The NR LDPC design [6] includes a raptor-like extension mechanism that can, in principle, generate unlimited parity bits (in practice constrained by the minimum mother code rate). This enables ﬁnegrained incremental redundancy while exploiting the quasi-cyclic structure of LDPC codes for efﬁcient implementation. In legacy systems such as WiFi, LDPC codes are deﬁned only for a ﬁnite set of ﬁxed code rates and block lengths. Each codeword is generated from a parity-check matrix tailored to a speciﬁc rate. This inherently limits the granularity of incremental redundancy and makes it difﬁcult to support truly rateless IR-HARQ.

ZHANG, WANG AND TONG: BEYOND 1→N DECODING: CAPACITY-AWARE RATELESS POLAR CODES FOR IR-HARQ

By contrast, 5G NR adopts a protograph-based raptorlike LDPC construction [8], as shown in Fig. 3. Each NR LDPC base graph contains: • A relatively high-density core (functionally similar to the outer codes of a Raptor code); • A raptor-like low-density extension part (analogous to the inner codes of a Raptor code) that can generate additional parity checks on demand. The key feature is that the extension parity bits are formed as linear combinations of (pseudo-)randomly chosen information bits. This directly realizes the rateless property: the encoder can, in principle, keep producing new, linearly independent parity bits without redesigning the code. This design has several important implications for IRHARQ: 1) Fine-grained rate adaptation: The raptor-like extension enables 1-bit (or very small step) granularity in rate adaptation. A higher-rate codeword is obtained by puncturing a lower-rate “mother” codeword; this nesting is inherent to the paritycheck matrix construction and does not require separate code designs for each target rate. 2) Single mother code, rate-independent construction: The LDPC code is effectively constructed once (via the base graph and its lifting) and is independent of the ﬁnal transmitted length. This is a natural ﬁt to IR-HARQ, where the code rate is not known a priori and adapts to the channel via feedback. 3) Near-uniform performance across a wide rate range: For well-designed base graphs (e.g., 5G NR BG1 and BG2), the same mother code delivers capacity-approaching performance over a broad range of effective code rates. That is, code rate ﬂexibility does not signiﬁcantly compromise performance. This is crucial for IR-HARQ, where decoding must remain close to optimal after each incremental redundancy transmission, not only at one or two predeﬁned rates. Thanks to these properties, 5G’s raptor-like LDPC codes can fully exploit the potential of rateless IRHARQ. III. A RE POLAR CODES RATELESS ? The quick answer is: conventional polar codes [9] cannot fully support rateless IR-HARQ. The fundamental obstacle comes from the length-dependent nature of polar code construction. This makes it impossible, in general, to build a single “mother” polar code from which all higher-rate versions are its nested codewords.

A. Length-dependent construction of polar codes A polar code [9] of length 𝑁 and dimension 𝐾 is constructed by selecting the 𝐾 most reliable polarized subchannels to carry information bits; the remaining 𝑁 − 𝐾 least reliable subchannels are frozen to known values. The reliability of these subchannels: • depends on the underlying physical channel, and • changes with both the code length and the ratematching operation (puncturing/shortening). Puncturing a code bit makes its corresponding bitchannel have effectively zero capacity; shortening yields absolute reliability (perfect a priori knowledge) while carrying no actual information. Consequently, different rate/length conﬁgurations lead to different reliability orderings and hence different information/frozen sets. Early work on rate-compatible polar codes [10], [11] used on-the-ﬂy reliability calculation that depends on instantaneous CSI (e.g., SNR), which is impractical in real systems because perfect, real-time CSI is unavailable. 5G polar codes [6] circumvent this by adopting an ofﬂine, channel-independent design: • A ﬁxed reliability sequence of length 1024 is precomputed; • Rate matching (via carefully designed puncturing and shortening patterns) attempts to preserve this pre-deﬁned reliability order for different lengths/rates. This provides practical rate and length ﬂexibility, but only at the level of designing a family of codes for different code rates - not a single mother code that is rate-compatible in the strict, rateless sense. B. Rate/length ﬂexibility ≠ rateless IR-HARQ Rateless IR-HARQ requires that: • A single mother code of length 𝑁 max and dimension 𝐾 is deﬁned. • In each transmission, a subset of the 𝑁 max coded bits are sent. For 5G polar codes, this nesting property does not hold. In practice: • High-rate codes employ shortening. • Low-rate codes employ puncturing. • The optimal puncturing and shortening patterns (and hence the optimal information sets) are not compatible for different rates. Thus, as the effective code rate decreases through incremental redundancy transmissions, the higher-rate code is not simply a subset of a lower-rate code deﬁned on the same underlying polar transform. This violates a key structural requirement for rateless IR-HARQ.

ZHANG, WANG AND TONG: BEYOND 1→N DECODING: CAPACITY-AWARE RATELESS POLAR CODES FOR IR-HARQ

Info bits (Iq)

Frozen bits

PC bits (Ip)

Info bits (Iq)

Frozen bits

Info bits

Bit copy

PC bits (Ip)

Bit vector to be encoded

Bit vector to be encoded 2nd Tx

Polar Encoding

2nd Tx

Polar Encoding

c'2

Polar Encoding

c'1

1st Tx

Fig. 4. Polar-based IR-HARQ via incremental freezing

Fig. 5. Polar-based IR-HARQ via polarizing matrix extension

C. Existing polar IR-HARQ schemes

This can, in principle, yield coding and diversity gains closer to those of an actual length-2𝑁 polar code. The scheme is illustrated in Fig. 5. Speciﬁcally, since polar code construction is length-dependent, we let: • I1 be the optimal information set (size 𝐾 ) for length 𝑁, • I2 be the optimal information set (size 𝐾 ) for length 2𝑁 . Typically, I1 ≠ I2 , which implies: • Some positions I𝑝 ⊂ I1 that are reliable at length 𝑁 become less reliable at length 2𝑁 , • Some new positions I𝑞 = I2 \ I1 become more reliable at length 2𝑁 . To retain optimality for both code length 𝑁 and 2𝑁 , one approach is to map the same information bits to positions in I𝑝 and I𝑞 , so that: • In the ﬁrst transmission (length 𝑁 ), I1 is used as the information set, giving the optimal length-𝑁 performance. • In the combined length-2𝑁 code, I2 is used, and the bits in I𝑝 can be treated as parity-check frozen bits, since their values have already been recovered from I𝑞 . This scheme improves over [12] by more closely matching a longer polar code. However, it still does not provide a fully ﬂexible rateless solution in general.

There are two important works on polar IR-HARQ, one based on incremental freezing [12] and the other is based on polarization matrix extension [13]. For clarity, we ﬁrst assume that there are two transmissions of length 𝑁 each. 1) Incremental freezing: The incremental freezing scheme is illustrated in Fig. 4 and works as follows: • In the ﬁrst transmission, a length- 𝑁 polar code of rate 𝑅1 = 𝐾/𝑁 is used. • In the second transmission, some unreliable bits from the ﬁrst transmission are re-encoded into a lower-rate polar code and retransmitted. • If they are successfully decoded, their values are regarded as known and frozen in a second decoding attempt of the ﬁrst codeword - effectively reducing the its code rate. This mechanism does exploit HARQ-type gains, but both transmissions are encoded and decoded as independent length-𝑁 polar codes. They are not jointly decoded as a single length-2𝑁 polar code. As a result, the scheme cannot harvest the full coding gain that would be available from a truly length-2𝑁 polar code. 2) Polarizing matrix extension: To approach the performance of longer codes, [13] proposes matrix extension: • The length- 𝑁 polar transform used in the ﬁrst transmission is extended to form a larger polar transform (e.g., to length 2𝑁 ). • After a retransmission, the two length- 𝑁 blocks are combined and jointly decoded as a length-2𝑁 polar code.

•

D. Fundamental limitations For potential applications in 6G and beyond [14], arbitrary transmissions length should be supported, and the

ZHANG, WANG AND TONG: BEYOND 1→N DECODING: CAPACITY-AWARE RATELESS POLAR CODES FOR IR-HARQ

inherent length-dependence of polar code construction leads to two critical limitations. 1) No universal mother code: For a truly rateless scheme, one desires a single mother code of length 𝑁max and dimension 𝐾 . For polar codes, this is unattainable in general because: • The optimal information set is tied to the actual code length and the rate matching pattern. • Polar codes require re-optimizing the information/frozen set whenever the effective code length/rate changes. Thus, there is no universal polar mother code with a rate-independent construction that supports arbitrary incremental redundancy in a strictly nested fashion. 2) Loss of coding gain and ﬂexibility for arbitrary retransmission lengths: Even with matrix extension [13], optimal coding gain can often be guaranteed only for speciﬁc retransmission lengths (e.g., two equal-length transmissions). For arbitrary retransmission lengths, serious issues arise. Consider an extreme example: • First transmission length: 𝐸 1 = 𝑁 . • Second transmission length: 𝐸 2 = 1. To embed this into a length-𝐸 (2) = 𝐸 1 + 𝐸 2 = 𝑁 + 1 polar transform via matrix extension, the extended part has 𝑁 − 1 punctured bits - they correspond to 𝑁 − 1 zero-capacity bit channels. The resulting polarized subchannels for the extended part are so unreliable that: • All the polarized subchannels associated with the extended part must be frozen. • These frozen bit positions cannot later be converted into information bit positions in further retransmissions. As such, any subsequent retransmissions will have no coding gain at all. Recall the basic 2 × 2 polar transform: (𝑢1 , 𝑢2 ) ↦→ (𝑐1 , 𝑐2 ) = (𝑢1 ⊕ 𝑢2 , 𝑢2 ).

If the upper-left input bit 𝑢1 is frozen, then 𝑐1 becomes a repetition of 𝑢2 . Now imagine 𝑁 such transforms in parallel. When all extended subchannels are frozen due to extreme puncturing, the new code bits corresponding to the retransmission are effectively repetitions of existing code bits from the initial transmission. They therefore contribute almost no additional coding gain; they mainly provide repetition gain rather than genuine new redundancy. This phenomenon is not limited to the pathological case 𝐸 2 = 1; it reveals that: • Once a bit is frozen due to length-dependent construction under severe puncturing/shortening, it cannot be “reclaimed” for carrying new information in subsequent transmissions.

Code Property Fixed (𝑁, 𝐾) Flexible (𝑁, 𝐾) Rateless (nested 𝑁)

RM Turbo LDPC Polar Ø Ø Ø Ø × Ø 3G/4G Ø 5G Ø 5G – 3G/4G Ø 5G This work × Ø TABLE I C ODE PROPERTY OF 3GPP STANDARDIZED CODES

•

Consequently, for many arbitrary IR patterns (variable 𝐸 1 , 𝐸 2 , 𝐸 3 , . . . ), the achievable coding gain is signiﬁcantly constrained, and in some cases, additional transmissions bring little or no coding beneﬁt.

Remark 1. Therefore, conventional polar codes whether in their 5G form or in existing IR-HARQ schemes - are not rateless. Achieving genuine rateless IR-HARQ with polar codes would require fundamentally new constructions that decouple code design from blocklength in a way that preserves nesting and maintains near-optimal performance at all cumulative lengths. Based on the discussions in Section I, II and III, we conclude that LDPC codes are rateless, but turbo codes and polar codes do not fully qualify as rateless, as summarized in Table I. The goal of this work is to provide the rateless property for polar codes. IV. P OLAR CODES : CHANNEL - DEPENDENT OR CHANNEL - INDEPENDENT ? In this section, we revisit a fundamental dilemma that traces back to the very deﬁnition of polar codes, and explain why this original deﬁnition seems to prevent them from being channel-adaptive and thus inherently rateless. The key issue is the apparent channel dependence of polar code construction in theory, versus the channel-independent constructions that are widely used in practice. A. Channel-dependent in theory In Arıkan’s seminal paper [9], a polar code of length 𝑁 and dimension 𝐾 is deﬁned by specifying the information set I ⊂ {1, . . . , 𝑁 } of size |I| = 𝐾 , such that the polarized subchannels indexed by I are the most reliable ones. Formally, I is chosen to satisfy 𝑗

𝑖 𝑍 (𝑊 𝑁 ) ≤ 𝑍 (𝑊 𝑁 ),

∀𝑖 ∈ I, 𝑗 ∈ Ī,

(1)

𝑖 is the 𝑖 -th polarized subchannel induced by where 𝑊 𝑁 the physical channel 𝑊 , and 𝑍 (·) denotes the Bhattacharyya parameter (a reliability metric). Since the set 𝑖 } and their reliabilities 𝑍 (𝑊 𝑖 ) depend explicitly on {𝑊 𝑁 𝑁 the underlying physical channel 𝑊 , Arıkan remarked that “Polar codes are channel-speciﬁc designs: a polar code for one channel may not be a polar code for another” [9].

ZHANG, WANG AND TONG: BEYOND 1→N DECODING: CAPACITY-AWARE RATELESS POLAR CODES FOR IR-HARQ

Early polar code construction schemes fully embraced this channel dependence. Classical works such as [15], [16], and [17] evaluate the reliability of the polarized subchannels with respect to a given channel 𝑊 and its SNR (or equivalent parameterization): • Density evolution (DE)-based methods [15], [16] track the full probability density functions of loglikelihood ratios (LLRs) through the polarization transform to obtain highly accurate reliability estimates. • Gaussian approximation (GA)-based methods [17] approximate the LLR distributions as Gaussian and propagate their means/variances, trading off some accuracy for much lower complexity. In all these cases, the construction explicitly requires a channel model and a channel parameter (e.g., SNR), and the resulting information set I is, by deﬁnition, channeldependent. B. Channel-independent in practice During 5G NR standardization (around 2015), it became clear that strict channel-dependent constructions pose serious implementation challenges: • Lack of perfect channel knowledge: Perfect and static channel state information is unavailable at the transmitter in practical wireless systems, and even the receiver only has imperfect, time-varying estimates. • Real-time complexity constraints: Even if perfect channel parameters were available, performing DE or GA online for each transport block or SNR operating point is prohibitively complex for hardware implementations, especially under stringent latency and power constraints. To overcome these issues, several channelindependent polar code construction methods were proposed and evaluated. A representative approach is the Polarization Weight (PW) method [18], which assigns to each polarized subchannel a reliability weight derived from its index using a simple, channelindependent “beta-expansion” formula [19]. In 5G NR, the ﬁnal solution is effectively a ﬁxed global reliability ordering: • A universal reliability sequence of length 1024 is speciﬁed in the standard, obtained ofﬂine from extensive simulations, analytical approximations, and machine-learning-based optimizations [20]. • For a given code length 𝑁 and dimension 𝐾 , the information set I is chosen as the indices corresponding to the 𝐾 most reliable positions in this pre-deﬁned sequence.

This design is channel-independent at run time: the transmitter and receiver simply use a look-up table. No channel model, no SNR parameter, and no on-theﬂy DE/GA computation are required. This guarantees interoperability and keeps complexity extremely low. Why do channel-independent constructions work in practice? The key observation is that practical systems operate near a speciﬁc “working region” of SNR and rate, not across all possible channel conditions. Consider a wireless system that adapts its modulation and coding scheme (MCS) so that the code rate 𝑅 = 𝐾/𝑁 roughly matches the available channel capacity. For example, a polar code of rate 𝑅 = 1/2 is not designed to operate extremely high SNR (e.g., 20 dB) or extremely low SNR (e.g., -20 dB). Instead, it has an intended operating SNR region, say around 0∼3 dB. Within such a working region: • The separation between “good” and “bad” subchannels, i.e., between I and Ī , becomes relatively stable and “deterministic”. • The reliability ordering within I (or within Ī ) is irrelevant since polar codes are deﬁned by I rather then the ordering within I . The above intuition can be turned into a conceptual procedure (Algorithm 1) for deriving a universal reliability sequence from any given channel-dependent construction method. Algorithm 1 Channel-independent reliability ordering 1: Input: Number of subchannels 𝑁 2: Output: Reliability ordered sequence 𝑄 3: 𝑄 ← [ ] {Initialize empty sequence} 4: for 𝑘 ← 1 to 𝑁 do 5: 𝑅 ← 𝑘/𝑁 {Target code rate} 6: SNR ← S (𝑅) {Operating SNR for rate 𝑅} 7: 𝑂 ← 𝑓 (𝑁, SNR) {Channel-dependent reliability order} 8: I ← 𝑂 [1 : 𝑘] {Top 𝑘 reliable positions} 9: 𝑖 ← I \ 𝑄 {Unique element in I not in 𝑄} 10: 𝑄 ← [ 𝑖, 𝑄 ] {Prepend new position} 11: end for 12: return 𝑄

C. The dilemma While the channel-dependent nature of polar codes prevents low-complexity ofﬂine construction, the channel-independent approach also fails to support nested codewords with a ﬁxed reliability ordering. Resolving this fundamental dilemma is essential for achieving rateless IR-HARQ.

ZHANG, WANG AND TONG: BEYOND 1→N DECODING: CAPACITY-AWARE RATELESS POLAR CODES FOR IR-HARQ

V. N EW POLARIZATION AND DECODING SCHEDULING In this section, we introduce a more general polarization framework enabled by an additional degree of freedom: the decoding schedule. Unlike the conventional setting, where the decoding order in successivecancellation (SC) decoding is ﬁxed, we explicitly regard the decoding schedule as a design parameter. This new framework allows code-length adaptation to be implemented at the decoder rather than the encoder, so that a single, ﬁxed code construction can achieve near-optimal performance over a wide range of effective code lengths. This framework paves the way toward rateless polar coding schemes. Classical channel polarization, starting from Arıkan’s seminal work and followed by most subsequent studies, assumes a ﬁxed “1 → 𝑁 ” sequential SC decoding schedule. Under this conventional paradigm, the reliability of the 𝑖 -th synthesized subchannel is deﬁned under the following implicit assumptions: 1) The SC decoding tree is traversed in a depth-ﬁrst manner, typically from the leftmost leaf to the rightmost leaf. 2) When making a hard decision on the 𝑖 -th bit, the previously decoded bits 1, 2, . . . , 𝑖 − 1 are assumed to have been decoded correctly. These assumptions are deeply embedded in the standard analysis of polarization, including the recursive evolution of subchannel capacity and reliabilities. If either of the above assumptions is relaxed, the evaluation methodology for polarization and the resulting subchannel reliabilities can change substantially.

A. Generalized single-step polar transform Recall that in Arıkan’s seminal work, the Bhattacharyya parameter 𝑍 (𝑊) is used to measure the reliability of a binary-input discrete memoryless channel (BDMC) 𝑊 , which provides an upper bound on the probability of a maximum-likelihood (ML) decision error. For the standard single-step polarization transform (𝑊, 𝑊) → (𝑊 ′ , 𝑊 ′′ ) under conventional decoding order (𝑊 ′ followed by 𝑊 ′′ ), the recursive relations are given by: 𝑍 (𝑊 ′ ) ≤ 2𝑍 (𝑊) − 𝑍 (𝑊) 2 ′′

𝑍 (𝑊 ) = 𝑍 (𝑊)

(2) (3)

where equalities hold if and only if 𝑊 is a Binary Erasure Channel (BEC).

W1 W2

y1 y2

Fig. 6. Generalized single-step polar transform

In the more general case where the input channels 𝑊1 and 𝑊2 are not statistically identical, the generalized single-step transform (𝑊1 , 𝑊2 ) → (𝑊 ′ , 𝑊 ′′ ) yields: 𝑍 (𝑊 ′ ) ≤ 𝑍 (𝑊1 ) + 𝑍 (𝑊2 ) − 𝑍 (𝑊1 )𝑍 (𝑊2 ) ′′

𝑍 (𝑊 ) = 𝑍 (𝑊1 )𝑍 (𝑊2 )

(4) (5)

Consistent with the homogeneous construction, the equality for 𝑍 (𝑊 ′ ) hold if and only if both 𝑊1 and 𝑊2 are BECs. Throughout this paper, we assume a BEC in the recursive calculation of Bhattacharyya parameters. For Additive White Gaussian Noise (AWGN) channels, the Bhattacharyya parameter calculated under the BEC assumption serves as an upper bound. This generalized recursion provides the analytical tool for evaluating subchannel reliability under arbitrary decoding schedules.

B. Code-length adaptation via decoding scheduling 1) An 𝑁 = 2 example: We ﬁrst consider the simplest generalized single-step transform (𝑊1 , 𝑊2 ) → (𝑊 ′ , 𝑊 ′′ ) , shown in Fig. 6. Let 𝑢1 and 𝑢2 denote the information bits. These are mapped to code bits 𝑥1 = 𝑢1 ⊕ 𝑢2 and 𝑥2 = 𝑢2 , which are then transmitted over channels 𝑊1 and 𝑊2 , resulting in the observations 𝑦 1 and 𝑦 2 , respectively. There can be two decoding orders: Scheduling 𝑢1 → 𝑢2 : This is the conventional decoding order, and the standard recursive relations (4), (5) hold. • Scheduling 𝑢 2 → 𝑢 1 : If we invert the schedule, we effectively treat 𝑢2 as the ﬁrst bit to be decoded. In this case, the transform effectively reduces to 𝑊2 → 𝑊 ′′ . Once 𝑢2 is decided, 𝑢1 = 𝑥1 ⊕ 𝑢2 effectively reduces to 𝑊1 → 𝑊 ′ . •

Under the reverse decoding order (𝑊 ′′ followed by 𝑊 ′ ), the recursive relations become: 𝑍 (𝑊 ′ ) = 𝑍 (𝑊1 ) ′′

𝑍 (𝑊 ) = 𝑍 (𝑊2 )

(6) (7)

ZHANG, WANG AND TONG: BEYOND 1→N DECODING: CAPACITY-AWARE RATELESS POLAR CODES FOR IR-HARQ

1 1

u2 u3 u4

2İ-İ2

İ2

•

Schedule 𝑆1 : Applying the recursive relations yields:

İ İ İ İ

𝑍5 (𝑢1 ) = 2𝜀 − 𝜀 2

3 4 7 89 10

𝑍5 (𝑢2 |𝑢1 ) = 𝜀 2 (2 − 𝜀) (1 + 𝜀 − 𝜀 2 )

𝑍5 (𝑢3 |𝑢1 , 𝑢2 ) = 𝜀 2 + 𝜀 3 − 𝜀 5

1 2 5 6

𝑍5 (𝑢4 |𝑢1 , 𝑢2 , 𝑢3 ) = 𝜀 5

•

Schedule 𝑆2 : Reordering the decoding sequence results in: 𝑍5 (𝑢2 ) = 𝜀 2 (2 − 𝜀) 2

Z5(u2)= İ2(2- İ)2; Z5(u3|u2)=2İ2-İ4; Z5(u4|u2,u3)= İ4; Z5(u1|u2,u3,u4)=İ

𝑍5 (𝑢3 |𝑢2 ) = 2𝜀 2 − 𝜀 4

𝑍5 (𝑢4 |𝑢2 , 𝑢3 ) = 𝜀 4

Fig. 7. An example of the 𝑢 2 → 𝑢 3 → 𝑢 4 → 𝑢 1 schedule and its recursive evolutions.

2) An (𝑁 ≥ 5, 𝐾 = 4) rateless polar coding example: To demonstrate the principle of code-length adaptation through decoding scheduling, we examine a rateless polar coding scenario with an information length of 𝐾 = 4. In this framework, the code length 𝑁 ∈ {5, 6, 7, 8, . . . } is not ﬁxed during construction or encoding, requiring the decoder to adapt to varying degrees of puncturing. For polar codes, a set of partial orders exists that reveals deterministic reliability relationships applicable to any binary-input memoryless symmetric channel (BMSC) [21]. For 𝑁 = {5, 6, 7, 8} under sequential puncturing, the subchannel reliability ordering is always: 1 2 3 5 𝑍 (𝑊 𝑁 ) ≥ 𝑍 (𝑊 𝑁 ) ≥ 𝑍 (𝑊 𝑁 ) ≥ 𝑍 (𝑊 𝑁 )

4 6 7 8 ≥ 𝑍 (𝑊 𝑁 ) ≥ 𝑍 (𝑊 𝑁 ) ≥ 𝑍 (𝑊 𝑁 ) ≥ 𝑍 (𝑊 𝑁 ).

To optimize performance for 𝐾 = 4, we allocate the information bits u = {𝑢1 , 𝑢2 , 𝑢3 , 𝑢4 } to the most reliable indices identiﬁed by the information set I = {4, 6, 7, 8}. We analyze two candidate decoding schedules: Schedule 1 (𝑆1 ): 𝑢1 → 𝑢2 → 𝑢3 → 𝑢4 . This is the standard schedule. Its recursive evolution follows (4), (5). • Schedule 2 (𝑆2 ): 𝑢 2 → 𝑢 3 → 𝑢 4 → 𝑢 1 . An example of this schedule is illustrated in Fig. 7 for 𝑁 = 5, 𝐾 = 4. Its recursive evolutions follow both (4), (5) and (6), (7), also shown in Fig. 7.

𝑍5 (𝑢1 |𝑢2 , 𝑢3 , 𝑢4 ) = 𝜀 •

Proof. Let Δ𝑍5 = 𝑃 𝐵 (𝑆1 ) − 𝑃 𝐵 (𝑆2 ) . Algebraic simpliﬁcation yields: Δ𝑍5 = − 𝜀(𝜀 − 1) 6 (𝜀 + 1)·

(𝜀 9 − 2𝜀 7 − 4𝜀 6 − 2𝜀 5 − 2𝜀 4 − 3𝜀 3 − 2𝜀 2 − 𝜀 − 1)

=( Negative) × ( Positive) × ( Positive) × ( Negative) >0.

Thus, 𝑆1 consistently yields a higher error probability than 𝑆2 at this code length. b) Case 2: Code Length 𝑁 = 6: With 𝑥1 , 𝑥2 punctured, the input reliabilities are 𝑍 (𝑥1 ) = 𝑍 (𝑥2 ) = 1 and 𝑍 (𝑥3 ) = · · · = 𝑍 (𝑥8 ) = 𝜀 . •

Schedule 𝑆1 :

𝑍6 (𝑢1 ) = (2𝜀 − 𝜀 2 ) 2

𝑍6 (𝑢2 |𝑢1 ) = (𝜀 + 𝜀 2 − 𝜀 3 ) 2

•

The performance is evaluated using the Bhattacharyya parameter 𝑍 (𝑊) to bound the block error probability Î𝐾 (1 − 𝑍 (𝑢𝑖 | . . . )) . 𝑃 𝐵 ≈ 1 − 𝑖=1 a) Case 1: Code Length 𝑁 = 5: With the ﬁrst three code bits punctured, the channel reliabilities are 𝑍 (𝑥1 ) = 𝑍 (𝑥2 ) = 𝑍 (𝑥3 ) = 1 and 𝑍 (𝑥4 ) = · · · = 𝑍 (𝑥8 ) = 𝜀 .

Comparison: For 𝑁 = 5, 𝑆2 is strictly superior for all 𝜀 ∈ (0, 1) .

𝑍6 (𝑢3 |𝑢1 , 𝑢2 ) = 2𝜀 3 − 𝜀 6

𝑍6 (𝑢4 |𝑢1 , 𝑢2 , 𝑢3 ) = 𝜀 6 •

Schedule 𝑆2 : 𝑍6 (𝑢2 ) = 𝜀 2 (2 − 𝜀) 2

𝑍6 (𝑢3 |𝑢2 ) = 2𝜀 2 − 𝜀 4

𝑍6 (𝑢4 |𝑢2 , 𝑢3 ) = 𝜀 4

𝑍6 (𝑢1 |𝑢2 , 𝑢3 , 𝑢4 ) = 𝜀 2 •

Comparison: At 𝑁 = 6, the optimal schedule becomes channel-dependent.

ZHANG, WANG AND TONG: BEYOND 1→N DECODING: CAPACITY-AWARE RATELESS POLAR CODES FOR IR-HARQ

Proof. 𝜀 2 (𝜀 − 1) 6 (𝜀 + 1) 2 (𝜀 2 − 2𝜀 − 1)·

(𝜀 10 − 4𝜀 7 − 2𝜀 6 − 4𝜀 5 − 2𝜀 3 + 𝜀 2 + 2)

10 -2

10 -3

=( Positive) × ( Positive) × ( Positive) × ( Negative) × 𝑓 (𝜀),

Detailed proofs for 𝑁 ∈ {7, 8} are provided in Appendix B, which similarly demonstrate channeldependent switching at 𝜀 𝑡 ℎ ≈ 0.920 and 0.965, respectively. Remark 2. This behavior indicates that polar codes exhibit channel-dependency not only in their construction (frozen set selection) but also in their decoding scheduling. For practical operating regions (𝜀 < 0.5), the optimal scheduling is primarily code-length-dependent, as shown in Table II. This dependency arises because different code lengths correspond to distinct puncturing patterns and varying distributions of zero-capacity input channels.

TABLE II O PTIMAL D ECODING S CHEDULE FOR 𝐾 = 4 N=5 × Ø

N=6 Ø ×

N=7 Ø ×

N=8 Ø ×

The practical implication is that by employing channel-aware decoding scheduling, we can maintain a ﬁxed code construction while adapting to ﬂuctuating channel conditions or code lengths to ensure optimal performance. Although the analysis of schedules 𝑆1 and 𝑆2 assumes a BEC, the code-length-dependent scheduling phenomenon is observed in AWGN channels as well, as demonstrated by the following simulation results. Fig. 8 presents block error rate (BLER) results for 𝐾 = 4, 𝑁 = 5, 6, 7, 8, which clearly demonstrates that the decoding order is a critical design parameter. In traditional polar codes, the decoding order is ﬁxed. However, by enabling ﬂexible decoding scheduling, we can achieve code-length adaptation.

EsN0(dB)

EsN0(dB) K=4,N=8

10 -1

10 -2

10 -3

10 -2

10 -3

BLER

The difference Δ𝑍6 is governed by the function 𝑓 (𝜀) . Since 𝑓 (0) = 2 and 𝑓 (1) = −8, there exists a unique root 𝜀 𝑡 ℎ ≈ 0.746. Consequently: – 𝑆1 is optimal for 0 < 𝜀 < 0.746. – 𝑆2 is optimal for 0.746 < 𝜀 < 1.

EsN0(dB) K=4,N=7

10 -1

K=4,N=6

10 -1

BLER

Δ𝑍6 = 𝑃 𝐵 (𝑆1 ) − 𝑃 𝐵 (𝑆2 ) =

Decoding Schedule 𝑢1 → 𝑢2 → 𝑢3 → 𝑢4 𝑢2 → 𝑢3 → 𝑢4 → 𝑢1

K=4,N=5

10 -1

10 -2

10 -3

EsN0(dB)

Fig. 8. Performance comparison of different decoding scheduling for (𝑁 ≥ 5, 𝐾 = 4).

3) SC decoding with arbitrary scheduling: The fundamental shift from 1 → 𝑁 sequential SC is the introduction of a decoding schedule S . Deﬁnition: A decoding schedule S = {𝑠1 , 𝑠2 , . . . , 𝑠 𝐾 } is an ordered permutation of the information indices in I that dictates the sequence in which leaf nodes are visited and hard-decided. The decoding schedule S introduces a degree of freedom to adapt to different channel conditions or code lengths, such as punctured codes, by prioritizing the decoding of subtrees with higher available mutual information. If the schedule dictates that 𝑢 𝑗 is decoded before 𝑢𝑖 (where 𝑗 > 𝑖 ), the decoder prioritizes the lower subtree of the corresponding polar transform. The decoder operates on LLRs. For any bit 𝑥 , the LLR is deﬁned as: 𝑃(𝑦|𝑥 = 0) 𝜆(𝑥) = ln (8) 𝑃(𝑦|𝑥 = 1) where 𝑦 is the channel observation. We denote 𝝀 𝑑 as the LLR vector at depth 𝑑 of the decoding tree, and 𝜷 𝑑 as the corresponding vector of hard-decision partial sums. We deﬁne the three primary processing functions: • 𝑓 -function: Soft estimate of 𝑢 1 when 𝑢 2 is unknown. 𝜆 𝑢1 = 𝑓 (𝜆 𝑥1 , 𝜆 𝑥2 ) 𝜆𝑥 𝜆𝑥 = 2 tanh −1 tanh 1 · tanh 2 2 2 • 𝑔 -function: Soft estimate of 𝑢 2 given 𝑢ˆ 1 .

(9)

𝜆 𝑢2 = 𝑔(𝜆 𝑥1 , 𝜆 𝑥2 , 𝑢ˆ1 ) = 𝜆 𝑥2 + (−1) 𝑢ˆ 1 𝜆 𝑥1

(10)

•

ℎ-function (newly introduced reverse cancellation): Soft estimate of 𝑢1 given 𝑢ˆ2 . 𝜆 𝑢1 = ℎ(𝜆 𝑥1 , 𝑢ˆ 2 ) = (−1) 𝑢ˆ 2 𝜆 𝑥1

(11)

ZHANG, WANG AND TONG: BEYOND 1→N DECODING: CAPACITY-AWARE RATELESS POLAR CODES FOR IR-HARQ

The whole decoding process is decomposed into these basic transforms, which are recursively processed according to the sequence deﬁned in S .

To handle arbitrary decoding orders while maintaining memory efﬁciency, we propose a generalized SC decoding framework in Algorithm 2.

Algorithm 2 Successive Cancellation Decoding with Arbitrary Schedule 1: Input: Channel LLRs 𝝀 𝑐ℎ , Info set I , Schedule S = {𝑠1 , . . . , 𝑠 𝐾 } 2: Output: Decoded vector û 3: û ← vector of size 𝑁 initialized to None 4: 𝑝 ← 1 {Global schedule pointer} 5: while 𝑝 ≤ 𝐾 do 6: if û[𝑠 𝑝 ] is None then 7: RecursiveDecode(𝑛, 1, 𝝀 𝑐ℎ ) 8: else 9: 𝑝 ← 𝑝+1 10: end if 11: end while 12: Function RecursiveDecode(𝑑, 𝑣, 𝝀) 13: if 𝑝 > 𝐾 OR 𝑠 𝑝 ∉ L (𝑑, 𝑣) then 14: goto line 6 {Exit subtree and return to root} 15: end if 16: if 𝑑 = 0 then 17: û[𝑣] ← (𝑣 ∈ I and 𝝀 < 0)?1 : 0 18: 𝑝 ← 𝑝+1 19: return û[𝑣] 20: end if 21: 𝑠𝑡 𝑎𝑟 𝑔𝑒𝑡 ← S [ 𝑝] 22: (𝝀1 , 𝝀2 ) ← Split 𝝀 into upper/lower halves 23: if 𝑠𝑡 𝑎𝑟 𝑔𝑒𝑡 ∈ L (𝑑 − 1, 2𝑣 − 1) then 24: {Standard Order: Upper subtree ﬁrst} 25: 𝝀𝑢 𝑝 ← 𝑓 (𝝀1 , 𝝀2 ) 26: 𝛽1 ← RecursiveDecode(𝑑 − 1, 2𝑣 − 1, 𝝀𝑢 𝑝 ) 27: 𝝀𝑙𝑜𝑤 ← 𝑔(𝝀1 , 𝝀2 , 𝛽1 ) 28: 𝛽2 ← RecursiveDecode(𝑑 − 1, 2𝑣, 𝝀𝑙𝑜𝑤 ) 29: else 30: {Reversed Order: Lower subtree ﬁrst} 31: 𝝀𝑙𝑜𝑤 ← 𝝀2 32: 𝛽2 ← RecursiveDecode(𝑑 − 1, 2𝑣, 𝝀𝑙𝑜𝑤 ) 33: 𝝀𝑢 𝑝 ← ℎ(𝝀1 , 𝛽2 ) 34: 𝛽1 ← RecursiveDecode(𝑑 − 1, 2𝑣 − 1, 𝝀𝑢 𝑝 ) 35: end if 36: return (𝛽1 ⊕ 𝛽2 , 𝛽2 ) {Propagate partial sums to parent}

In this algorithm, we deﬁne L (𝑑, 𝑣) as the set of leaf indices descendant from node 𝑣 at depth 𝑑 . For a polar

code of length 𝑁 = 2𝑛 , this is deﬁned as: L (𝑑, 𝑣) = 𝑖 ∈ Z : (𝑣 − 1)2𝑑 + 1 ≤ 𝑖 ≤ 𝑣2𝑑 .

When the next bit in the schedule 𝑠 𝑝 falls outside the current subtree L (𝑑, 𝑣) , the algorithm executes a global jump (Line 14) that immediately terminates the recursion stack and unwinds to the root (Line 6). This “break” mechanism allows the decoder to re-enter the trellis for the new target index. Because the decision vector û is persistent, previously computed hard decisions are preserved and utilized in subsequent decoding. The proposed arbitrary scheduling framework can be naturally extended to Successive Cancellation List (SCL) decoding [22], [23]. In the SCL variant, the global decision vector û is replaced by a set of 𝐿 path candidates. When the RecursiveDecode function reaches a leaf node (𝑑 = 0) in the information set I , each surviving path splits into two (representing 𝑢𝑖 = 0 and 𝑢𝑖 = 1), and the 𝐿 most likely paths are retained based on their Path Metrics (PM). 4) Channel-aware decoding scheduling: Given the generalized SC framework that supports arbitrary decoding orders, the optimization problem of interest is to determine the decoding schedule S that minimizes the block error rate for a speciﬁc channel condition or puncturing pattern. Formally, let P (I) denote the set of all 𝐾! permutations of the information set I . The optimal schedule S ∗ is deﬁned as: S ∗ = arg min 𝑃𝑒 (S). S∈ P (I)

(12)

While an exhaustive search of 𝐾! permutations is computationally prohibitive, we propose a sub-optimal yet computationally efﬁcient greedy approach. The proposed scheduling algorithm prioritizes the next bit to be decoded as the most critical one. This prioritization is necessitated by the inherent nature of sequential decoding: if the next bit is decoded incorrectly, the entire block is rendered erroneous. By addressing this bottleneck - ensuring the most reliable bit is decided ﬁrst - the algorithm maximizes the probability of a successful decoding trajectory. The reliability of each information subchannel is characterized by its bit error probability (or Bhattacharyya parameter) 𝑍 . Unlike conventional natural index ordered decoding, where the reliability of the 𝑘 -th bit is evaluated assuming bits 𝑢1 , . . . , 𝑢 𝑘−1 are known, our framework evaluates the error probability of 𝑢 𝑘 conditioned on the set of decided bits 𝑢 𝑠1 , . . . , 𝑢 𝑠𝑘−1 . The recursive density evolution deﬁned in (4)–(7) is used to track these reliabilities. The greedy scheduling procedure is formalized in Algorithm 3.

ZHANG, WANG AND TONG: BEYOND 1→N DECODING: CAPACITY-AWARE RATELESS POLAR CODES FOR IR-HARQ

To demonstrate the efﬁcacy of the greedy approach, we revisit the example with 𝐾 = 4, 𝑁 ∈ {5, 6, 7, 8}, representing different puncturing patterns and channel conditions. Case 𝑁 = 5: The unconditional error probability 𝑍5 (𝑢2 ) = 𝜖 2 (2 − 𝜖) 2 is the minimum among the set {𝑢1 , 𝑢2 , 𝑢3 , 𝑢4 }. Thus, 𝑢2 is scheduled ﬁrst. Conditioned on 𝑢2 , the lowest reliability belongs to 𝑢3 where 𝑍5 (𝑢3 |𝑢2 ) = 2𝜖 2 − 𝜖 4 . Finally, comparing the remaining bits, we ﬁnd 𝑍5 (𝑢4 |𝑢2 , 𝑢3 ) = 𝜖 4 is lower than that of 𝑢1 . The resulting optimal greedy schedule is S = {𝑢2 , 𝑢3 , 𝑢4 , 𝑢1 }. • Case 𝑁 = 6: A tie occurs between 𝑢 1 and 𝑢 2 as 𝑍6 (𝑢1 ) = 𝑍6 (𝑢2 ) = 𝜖 2 (2 − 𝜖) 2 . To break the tie, we “look ahead” to evaluate the next bits 𝑍6 (𝑢2 |𝑢1 ) = (𝜖+𝜖 2 −𝜖 3 ) 2 while 𝑍6 (𝑢3 |𝑢2 ) = 2𝜖 2 −𝜖 4 . Since 𝑍6 (𝑢2 |𝑢1 ) < 𝑍6 (𝑢3 |𝑢2 ) for all 𝜖 ∈ (0, 1) , the algorithm selects 𝑢1 followed by 𝑢2 . The complete schedule is determined as S = {𝑢1 , 𝑢2 , 𝑢3 , 𝑢4 }. • Case 𝑁 ∈ {7, 8}: For higher values of 𝑁 , the channel provides sufﬁcient redundancy for 𝑢1 to become the most reliable bit, with 𝑍7 (𝑢1 ) = (2𝜖 − 𝜖 2 ) 3 and 𝑍8 (𝑢1 ) = (2𝜖 − 𝜖 2 ) 4 . Both metrics are superior to 𝑍 (𝑢2 ) in their respective cases, causing Algorithm 3 to converge to the standard sequential schedule S = {𝑢1 , 𝑢2 , 𝑢3 , 𝑢4 }.

•

These results corroborate the analytical ﬁnding in Section V-B2 that for heavily punctured codes, the optimal decoding schedule may deviate from the natural index order. Although Algorithm 3 assumes a BEC, its heuristic applies to other channels as well. For AWGN channels, the conditional error probability in Step 7 can be calculated through density evolution or Gaussian approximation. The simulation results in Section VI-E demonstrate its efﬁcacy in AWGN channels.

Decoding Schedule (S) New degree of freedom

Algorithm 3 Channel-Aware Decoding Scheduling 1: Input: Information set I , Channel condition 𝝐 2: Output: Decoding schedule S 3: S ← () {Initialize as an empty ordered sequence} 4: U ← I {Initialize set of unscheduled bits} 5: while U is not empty do 6: {Find the bit with the lowest conditional error probability} 7: 𝑠∗ ← arg min𝑘 ∈ U 𝑍 𝑁 (𝑢 𝑘 |S) 8: Append 𝑠∗ to S 9: U ← U \ {𝑠∗ } 10: end while 11: return S

2-dimensional design space

Traditional 1D optimization space

Code Construction (Information Set I)

Fig. 9. Expanding the Design Space: Code Construction (I) × Decoding Schedule (S)

5) New degree of freedom: The fundamental essence of this contribution lies in the introduction of a new degree of freedom within the polar code design space, as illustrated in Fig. 9. Traditionally, for a given set of channel statistics or a speciﬁc rate-matching scheme, the optimization of polar codes was conﬁned to code construction - speciﬁcally the selection of information and frozen sets. By formalizing the decoding schedule as an independent design variable, we transition from a one-dimensional optimization problem to a dual-degree-of-freedom framework. VI. F LEXIBLE IR-HARQ IN W IRELESS C OMMUNICATIONS In this section, we apply the proposed arbitrary decoding schedule to the IR-HARQ problem. By synergistically combining a ﬁxed nested code construction with dynamic scheduling, polar codes can seamlessly adapt to arbitrary block lengths. A. Nested Code Construction for Arbitrary Lengths To achieve ratelessness, the transmitted code length 𝐸 must be allowed to vary continuously between a minimum mother code length 𝑁min and a maximum length 𝑁max . We achieve this by applying our decoding scheduling framework to a nested polar code construction. The core mechanism of this construction is a “oneto-many” information bit mapping, where a single information bit is repeated (or copied) across multiple polarized subchannels in different subblocks. This introduces parity-check equations across the subblocks. The construction process is detailed in Algorithm 4.

ZHANG, WANG AND TONG: BEYOND 1→N DECODING: CAPACITY-AWARE RATELESS POLAR CODES FOR IR-HARQ

Algorithm 4 Nested Polar Code Construction 1: Input: Minimum length 𝑁 min , Maximum length 𝑁max , Information length 𝐾 2: Output: Global information set Imax , Mapping relations 3: 𝑁 ← 𝑁 min 4: I ← ∅, Imax ← ∅ 5: while 𝑁 ≤ 𝑁 max do 6: Construct optimal information set I ′ for PolarCode(𝑁, 𝐾 ) 7: I𝑝 ← I \ I ′ {Bits unique to previous block} 8: I𝑞 ← I ′ \ I {New bits in current block} 9: 𝑢 I𝑞 ← 𝑢 I𝑝 {Establish bit-copy mapping (detailed in Sec. VI-D)} 10: I ← I′ 11: Imax ← Imax ∪ I 12: 𝑁 ← 𝑁 ×2 13: end while 14: return Imax With the encoding structure ﬁxed, the remaining challenge is to design a decoding algorithm that optimizes performance for any arbitrary transmitted length 𝐸 . Speciﬁcally, we must adapt the generalized SC decoding (Algorithm 2) to leverage the parity-check relationships generated by the one-to-many bit mapping. B. Parity-Check Successive Cancellation Decoding with Arbitrary Schedule The “bit copy” operations in Algorithm 4 mean that an information bit exists in multiple bit channels across different subblocks. This creates an explicit parity-check relationship: if a bit 𝑢𝑖 is copied to 𝑢 𝑗 , then 𝑢ˆ 𝑖 = 𝑢ˆ 𝑗 . Consequently, if the schedule dictates that 𝑢𝑖 is decoded before 𝑢 𝑗 , the decision 𝑢ˆ𝑖 instantly determines 𝑢 𝑗 . The bit 𝑢 𝑗 effectively transitions from an unknown information bit to a known frozen bit for the remainder of the decoding process. We describe the adapted version of the generalized SC algorithm in a concise form (Algorithm 5), highlighting only the functional modiﬁcations while omitting unchanged components. Speciﬁcally, we introduce a tracking mechanism that updates all associated bit copies the moment a decision is made at the leaf node (𝑑 = 0). C. Parity-Check Channel-Aware Decoding Scheduling The channel-aware scheduling algorithm is also adapted to account for the parity-check relationships deﬁned by the copy-sets C . This adaptation is formalized in Algorithm 6.

Algorithm 5 Parity-Check SC Decoding with Arbitrary Schedule (Concise) 1: Input: Channel LLRs 𝝀𝑐ℎ , Info set I , Schedule S , Copy-sets C 2: Output: Decoded vector û 3: {Initialization and main WHILE loop remain identical to Algorithm 2} 4: . . . 5: Function RecursiveDecode(𝑑, 𝑣, 𝝀) 6: if 𝑝 > 𝐾 OR 𝑠 𝑝 ∉ L (𝑑, 𝑣) then 7: goto Main Loop {Exit subtree and return to root} 8: end if 9: if 𝑑 = 0 then 10: 𝑢ˆ 𝑡𝑒𝑚 𝑝 ← (𝝀 < 0)?1 : 0 11: for all 𝑤 ∈ CopySet (𝑣, C) do 12: û[𝑤] ← 𝑢ˆ 𝑡𝑒𝑚 𝑝 {Instantly freeze all copied bits} 13: end for 14: 𝑝 ← 𝑝+1 15: return 𝑢ˆ 𝑡𝑒𝑚 𝑝 16: end if 17: {Standard/Reversed recursive LLR calculations remain identical to Algorithm 2} 18: . . . 19: return (𝛽1 ⊕ 𝛽2 , 𝛽2 ) Once a bit 𝑠∗ is selected by the scheduler, all coupled bits in its associated copy-set are immediately appended to the scheduling sequence S (Lines 11–17). This mechanism ensures that these bits are treated as known frozen bits in all subsequent reliability evaluations, 𝑍 𝑁 (𝑢 𝑘 |S) , effectively reducing the code rates of the corresponding subblocks. D. Dynamic Interplay: Information Bit Mapping and Decoding Scheduling A pivotal feature of this framework is the dynamic synergy between the one-to-many information mapping and the arbitrary SC schedule. This interplay is best understood through the following conceptual categorization of subblocks: • Capacity-sufﬁcient: A subblock (typically the mother block) that is fully transmitted with relatively low effective code rate. • Capacity-deﬁcient: A subblock (typically the extension block) that is severely punctured, resulting in an effective code rate that exceeds the channel capacity. Because Algorithm 5 instantly freezes copied bits across all subblocks, the effective code rate of each sub-

ZHANG, WANG AND TONG: BEYOND 1→N DECODING: CAPACITY-AWARE RATELESS POLAR CODES FOR IR-HARQ

Algorithm 6 Channel-Aware Scheduling with Parity Checks 1: Input: Information set I , Channel condition 𝝐 , Copy-sets C 2: Output: Parity-aware decoding schedule S 3: S ← () {Initialize as an empty ordered sequence} 4: U ← I {Initialize set of unscheduled bits} 5: while U is not empty do 6: {Find the bit with the lowest conditional error probability} 7: 𝑠∗ ← arg min𝑘 ∈ U 𝑍 𝑁 (𝑢 𝑘 |S) 8: {Append the selected bit and remove from unscheduled set} 9: Append 𝑠∗ to S 10: U ← U \ {𝑠∗ } 11: {Immediately resolve all associated bit copies} 12: for all 𝑤 ∈ CopySet (𝑠∗ , C) do 13: if 𝑤 ∈ U then 14: Append 𝑤 to S 15: U ← U \ {𝑤} 16: end if 17: end for 18: end while 19: return S block is not static. As the decoder progresses, a capacitydeﬁcient subblock can dynamically become capacitysufﬁcient. The scheduling algorithm ensures that a capacitysufﬁcient subblock is decoded ﬁrst. Once decoded, its decisions assist the capacity-deﬁcient subblocks through freezing the copied bits in those subblocks, as shown in Fig. 10. However, to maximize this assistive effect across arbitrary truncations of 𝐸 , the speciﬁc mapping order of 𝑢 I𝑞 = 𝑢 I𝑝 is critical. Assume the mother subblock (containing I𝑝 ) is fully transmitted, while the extension subblock (containing I𝑞 ) is partially transmitted due to puncturing. Since the transmitted length 𝐸 can be any value, we implement a reverse mapping strategy: the earliest decoded bits in I𝑝 are mapped to the earliest transmitted bits in I𝑞 . For a code of length 2𝑁 with sequential puncturing (indices 1, . . . , 2𝑁 − 𝐸 are punctured), the initial transmission uses indices 𝑁 + 1, . . . , 2𝑁 . The retransmission uses indices starting from 𝑁 counting backwards (𝑁, 𝑁 −1, . . . , 2𝑁 −𝐸 +1). The mapping rule is formalized in Algorithm 7. This reverse mapping guarantees that for any transmitted length 𝐸 ∈ [𝑁 + 1, 2𝑁] , the capacity-deﬁcient extension block beneﬁts from the earliest decoded bits

Decode first

Decode later Capacity-deficient block

Capacity-sufficient block

Information mapping: u1=u2

Punctured

Transmitted

Fig. 10. Conceptual categorization of subblocks in terms of rate and capacity.

Algorithm 7 Reverse Information Bit Mapping for Rateless IR-HARQ Require: Sets I𝑝 and I𝑞 from nested construction Ensure: Bit mapping pairs 1: I𝑝sort ← Sort I𝑝 by ascending bit index 2: I𝑞sort ← Sort I𝑞 by descending bit index {Aligns with sequential puncturing} 3: for 𝑖 = 1 to |I𝑝 | do 4: Map information bit: 𝑢 I𝑞sort [𝑖] ← 𝑢 I𝑝sort [𝑖] 5: end for of the mother block. Speciﬁcally, the ﬁrst decoded bit 𝑢 𝑠1 in the mother block reduces the effective code rate of the extension block regardless of 𝐸 , because the mapping 𝑢 𝑠2 ← 𝑢 𝑠1 ensures the target index 𝑠2 = 𝑁 is always included in the set of transmitted indices {𝑁, 𝑁 − 1, . . . , 2𝑁 − 𝐸 + 1}. Remark 3. The primary advantage of this dynamic scheduling mechanism is that the algorithm identiﬁes capacity-deﬁcient subblocks as transient bottlenecks. Rather than attempting to decode it prematurely, the decoder waits until the coupled bits in the capacitysufﬁcient subblock are conﬁdently decided. This ensures an adaptive, on-the-ﬂy matching between rate and capacity. E. Simulation results In this section, we evaluate the performance of the proposed rateless polar codes, implemented using the integrated framework of nested construction, paritycheck-adapted scheduling and decoding, and reverse bit mapping (Algorithms 4–7).

ZHANG, WANG AND TONG: BEYOND 1→N DECODING: CAPACITY-AWARE RATELESS POLAR CODES FOR IR-HARQ

K=448, Nmin=512 ~ Nmax=1024

CC-HARQ IR-HARQ (rateless) Fixed-length (non-rateless)

2dB gain

Es/N0@1e-2

2.5dB gain

2.6dB gain

2 Initial trans.

1st retrans.

2nd retrans.

3rd retrans.

RV1 N=680

RV2 N=800

RV3 N=950

RV0 N=520

550

600

650

700

750 N

800

850

900

950

1000

Fig. 11. Required 𝐸 𝑠 /𝑁0 to achieve BLER = 10−2 for the proposed rateless polar code (𝐾 = 448) across continuous code lengths 𝑁 ∈ [512, 1024].

We ﬁrst examine an exemplary scenario with information length 𝐾 = 448 (including a 16-bit CRC) and a mother code range of 𝑁min = 512 to 𝑁max = 1024. The actual transmitted block length 𝑁 varies continuously within this interval. Decoding is performed using CRCaided SCL decoding with a list size 𝐿 = 8. The performance metric is the required SNR (𝐸 𝑠 /𝑁0 ) to achieve a target BLER of 10−2 . As illustrated in Fig. 11, we compare our proposed scheme against two distinct benchmarks: • Chase Combining HARQ (CC-HARQ): A baseline where redundancy is achieved via simple bit repetition. This scheme provides energy gain but lacks additional coding gain, serving as a lower bound for performance. • Fixed-Length QUP Polar Codes: A benchmark where a unique polar code is constructed and encoded for each speciﬁc 𝑁 using Quasi-Uniform Puncturing (QUP) [10]. Since these codes are optimized for a single rate and are not restricted by nested constraints, they represent the performance upper bound. To emulate a realistic HARQ process, we examine four speciﬁc redundancy versions (RVs) at 𝑁 ∈ {530, 680, 800, 950}. Our observations are as follows: 1) Signiﬁcant Coding Gain: Compared to CCHARQ, the proposed scheme achieves gains of 0.2 dB, 2.0 dB, 2.5 dB, and 2.6 dB at the respective RVs. 2) Optimality: Remarkably, the proposed rateless codes achieve similar performance to the ﬁxed-

length QUP codes across the entire range. This indicates that the nested constraints do not incur a performance penalty when adaptive decoding schedule is employed. 3) Monotonicity: The 𝑁 -to-SNR curve is smooth and monotonically decreasing. This behavior is desirable for practical base station scheduling, as it ensures that additional redundancy consistently translates into predictable reliability gains. To verify the framework’s robustness, we simulated ﬁne-grained performance in a broader range of code rates and lengths, speciﬁcally 𝐾 ∈ {210, . . . , 870} with 𝑁min = 1024 and 𝑁max = 2048. As shown in Fig. 12, across all simulated rate-length combinations, the proposed scheme consistently matches the performance of independently designed, non-rateless polar codes. F. Hardware implementation results To evaluate the physical feasibility and hardware overhead of the proposed framework, we developed an ASIC (Application-Speciﬁc Integrated Circuit) implementation comparing the rateless polar decoder against a conventional polar decoder architecture. To ensure a fair comparison, both decoders are designed with a list size of 𝐿 = 8, support a maximum code length of 𝑁max = 1024, and were synthesized using the same process node. The physical layouts of both decoders are illustrated in Fig. 13. For the purposes of comparison, the chip area of the conventional polar decoder is normalized to 1.00 × 1.20 units. Under the same scaling, the proposed rateless polar decoder occupies a footprint of 1.11 × 1.33 units, representing an area overhead of approximately 23%. This modest increase is primarily attributed to the additional control logic and memory required to manage the arbitrary scheduling and parity-check constraints across the decoding list. The energy efﬁciency of the two designs was evaluated at a decoding clock frequency of 1 GHz. Power consumption measurements for a 𝐾 = 256, 𝑁 = 768 polar code indicate that the rateless polar decoder consumes approximately 1.22× the power of the conventional baseline. This 22% power increment is associated with the increased switching activity in the scheduling logic necessitated by providing a uniﬁed solution for nested code lengths. VII. D ISCUSSIONS A. Related works Adapting polar codes for HARQ systems is fundamentally challenging due to their rigid code-length

ZHANG, WANG AND TONG: BEYOND 1→N DECODING: CAPACITY-AWARE RATELESS POLAR CODES FOR IR-HARQ

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constraints. While various schemes have been proposed in the literature, they generally fall short of providing a fully ﬂexible, high-performance rateless solution. Early attempts to implement HARQ for polar codes often relied on CC-HARQ or the transmission of nonpolar-coded bits. For instance, the authors in [28] proposed an improved CC-HARQ scheme based on the equivalent puncturing patterns (EPPs) to complement the LLRs of punctured bits. To introduce incremental redundancy, the scheme in [26] selectively transmits uncoded information bits as redundancy. This concept was

later generalized in [27] by transmitting intermediate bits within the SC trellis. However, because the additionally transmitted bits in these schemes are not fully protected by the extended polar transformation, they fail to enjoy the full coding gain of polar codes. To achieve optimal coding gain, subsequent research focused on structurally extending the polar code itself. For practical ﬁnite block lengths, IR-HARQ schemes based on polarizing matrix extension [13], [24], [25] generally outperform IF-HARQ by providing extra coding gain. The “copy bits pair” mechanism was proposed in [24], where the value of a bit in the previous transmission is copied to its corresponding paired bit in the extension block. This concept was also combined with quasi-uniform puncturing (QUP) in [25]. Despite their improved ﬁnite-block-length performance, these matrix extension schemes suffer from severe structural limitations. Speciﬁcally, the scheme in [24] strictly requires the length of the retransmitted blocks to grow in multiples of two (i.e., the total transmitted length must be a power of two). Similarly, [25] is mathematically equivalent to [24] when the incremental lengths follow rigid power-of-two constraints. Even the adaptive extension scheme in [13], which we extensively analyzed in previous sections, lacks a uniﬁed decoding scheduling mechanism to gracefully handle arbitrary transmission lengths. Consequently, none of the existing literature provides

ZHANG, WANG AND TONG: BEYOND 1→N DECODING: CAPACITY-AWARE RATELESS POLAR CODES FOR IR-HARQ

a seamless, rateless IR-HARQ framework that maintains near-optimal coding gain for completely arbitrary and continuous code lengths, which motivates the joint codeconstruction and scheduling design proposed in this paper. B. Open problems The framework presented in this paper introduces a new multi-dimensional design space for polar codes, as illustrated by the additional degrees of freedom in Fig. 9. While we have utilized greedy scheduling (Algorithm 6) and heuristic bit mapping (Algorithm 7) for their practical efﬁciency, several fundamental questions remain open regarding the theoretical limits and architectural optimizations of this approach. A primary open question is the characterization of the jointly optimal code construction and decoding schedule. While our greedy approach yields near-optimal coding gains for ﬁnite block lengths, a rigorous informationtheoretic analysis is required to determine the optimal joint code construction and scheduling strategy that minimizes the required SNR across all possible lengths 𝐸. Hardware implementation of the proposed framework for high throughput remains a valuable research direction. Standard polar decoding enhancements, such as simpliﬁed SC (SSC) [29] and Fast-SSCL [30], rely on parallelizing speciﬁc node within an SC tree. Extending these techniques to support the arbitrary schedules and dynamic parity-check constraints of our framework is non-trivial but essential. Notably, efﬁcient hardware architectures have been developed for existing matrixextension IR-HARQ schemes in [31], [32]. Generalizing these architectures for arbitrary decoding orders and dynamic scheduling offers a pathway toward highthroughput rateless polar coding. From an application perspective, the channel-aware adaptivity of these rateless polar codes provides a robust mechanism against fading and interference. In such scenarios, speciﬁc code bits are subjected to deep fading or erasure. Our framework allows the decoder to strategically circumvent these unreliable bit positions by reordering the decoding schedule. A thorough investigation of this capability under standardized fading models (e.g., Rayleigh or Rician) and practical 3GPP preemption patterns could provide valuable insights into designing more resilient communication links for mission-critical applications. VIII. C ONCLUSION In this paper, we have presented a uniﬁed framework for rateless polar codes by introducing a new degree of

freedom in the design space, i.e., the joint optimization of nested code construction and decoding scheduling. The framework supports arbitrary and nested code lengths 𝐸 ∈ [𝑁min , 𝑁max ] . Simulation results demonstrate that the proposed rateless codes achieve near-optimal coding gains, matching the performance of independently optimized, ﬁxed-rate benchmarks across a wide range of rates and lengths. Moreover, an ASIC implementation for 𝐿 = 8 and 𝑁max = 1024 conﬁrms hardware feasibility, offering a robust, channel-aware solution for practical implementations. A PPENDIX A P ROOF FOR THE GENERALIZED SINGLE - STEP TRANSFORM

Consider two independent binary-input discrete memoryless channels (B-DMCs) 𝑊1 : X → Y1 and 𝑊2 : X → Y2 . The single-step polarization transform (𝑊1 , 𝑊2 ) → (𝑊 ′ , 𝑊 ′′ ) is deﬁned via the mapping 𝑋1 = 𝑈1 ⊕ 𝑈2 and 𝑋2 = 𝑈2 . The synthesized transition probabilities are: Õ 1 𝑊 ′ (𝑦 1 , 𝑦 2 |𝑢1 ) = 𝑊1 (𝑦 1 |𝑢1 ⊕ 𝑢2 )𝑊2 (𝑦 2 |𝑢2 ) 2 𝑢2 ∈ {0,1} (13) 1 𝑊 ′′ (𝑦 1 , 𝑦 2 , 𝑢1 |𝑢2 ) = 𝑊1 (𝑦 1 |𝑢1 ⊕ 𝑢2 )𝑊2 (𝑦 2 |𝑢2 ) (14) 2 A. Symmetric Capacity Conservation By the chain rule of mutual information and the fact that (𝑈1 , 𝑈2 ) → (𝑋1 , 𝑋2 ) is a bijection: 𝐼 (𝑊1 ) + 𝐼 (𝑊2 ) = 𝐼 (𝑋1 ; 𝑌1 ) + 𝐼 (𝑋2 ; 𝑌2 ) = 𝐼 (𝑋1 , 𝑋2 ; 𝑌1 , 𝑌2 )

= 𝐼 (𝑈1 , 𝑈2 ; 𝑌1 , 𝑌2 )

= 𝐼 (𝑈1 ; 𝑌1 , 𝑌2 ) + 𝐼 (𝑈2 ; 𝑌1 , 𝑌2 |𝑈1 )

= 𝐼 (𝑊 ′ ) + 𝐼 (𝑊 ′′ )

Note: For the speciﬁc case where 𝑊1 , 𝑊2 are Binary Erasure Channels (BECs), 𝐼 (𝑊 ′ ) = 𝐼 (𝑊1 )𝐼 (𝑊2 ) and 𝐼 (𝑊 ′′ ) = 𝐼 (𝑊1 ) + 𝐼 (𝑊2 ) − 𝐼 (𝑊1 )𝐼 (𝑊2 ) . B. Reliability of the Enhanced Channel 𝑊 ′′ The Bhattacharyya parameter 𝑍 (𝑊 ′′ ) is deﬁned as: Õ p 𝑊 ′′ (𝑦 1 , 𝑦 2 , 𝑢1 |0)𝑊 ′′ (𝑦 1 , 𝑦 2 , 𝑢1 |1) 𝑍 (𝑊 ′′ ) = 𝑦1 ,𝑦2 ,𝑢1

Substituting the deﬁnition of 𝑊 ′′ , we obtain 𝑍 (𝑊 ′′ ) = 𝑍 (𝑊1 )𝑍 (𝑊2) (see detailed derivation in (15)).

ZHANG, WANG AND TONG: BEYOND 1→N DECODING: CAPACITY-AWARE RATELESS POLAR CODES FOR IR-HARQ

′′

1 1 𝑊1 (𝑦 1 |𝑢1 )𝑊2 (𝑦 2 |0) · 𝑊1 (𝑦 1 |𝑢1 ⊕ 1)𝑊2 (𝑦 2 |1) 2 2 𝑦1 ,𝑦2 ,𝑢1 ! ! Õp © Õ 1ª Õ p = 𝑊1 (𝑦 1 |0)𝑊1 (𝑦 1 |1) 𝑊2 (𝑦 2 |0)𝑊2 (𝑦 2 |1) ® 2 𝑦 𝑦 𝑢 ∈ {0,1} 1 2 ¬ «1 = 1 · 𝑍 (𝑊1 ) · 𝑍 (𝑊2 ) = 𝑍 (𝑊1 )𝑍 (𝑊2)

𝑍 (𝑊 ) =

𝑍 (𝑊 ′ ) ≤

Õ 1h

(15)

i √ √ √ (𝑎 + 𝑏) 𝑐𝑑 + (𝑐 + 𝑑) 𝑎𝑏 − 2 𝑎𝑏𝑐𝑑

2 Õ√ Õ√ Õ√ Õ√ 1Õ 1Õ (𝑎 + 𝑏) (𝑐 + 𝑑) 𝑐𝑑 + 𝑎𝑏 − 𝑎𝑏 𝑐𝑑 = 2 𝑦 2 𝑦 𝑦 𝑦 𝑦 𝑦 𝑦1 ,𝑦2

= 1 · 𝑍 (𝑊2) + 1 · 𝑍 (𝑊1 ) − 𝑍 (𝑊1 )𝑍 (𝑊2) = 𝑍 (𝑊1 ) + 𝑍 (𝑊2 ) − 𝑍 (𝑊1 )𝑍 (𝑊2 )

C. Upper Bound for the Degraded Channel 𝑊 ′ Let 𝑎 = 𝑊1 (𝑦 1 |0), 𝑏 = 𝑊1 (𝑦 1 |1), 𝑐 = 𝑊2 (𝑦 2 |0), 𝑑 = 𝑊2 (𝑦 2 |1) . ′

𝑍 (𝑊 ) =

Õ 1p

𝑦1 ,𝑦2

(𝑎𝑐 + 𝑏𝑑) (𝑎𝑑 + 𝑏𝑐)

√

Using the inequality 𝐴2 + 𝐵2 √ − 𝐶 2 ≤ 𝐴 + 𝐵 − 𝐶√where √ 𝐴 = (𝑎 + 𝑏) 𝑐𝑑 , 𝐵 = (𝑐 + 𝑑) 𝑎𝑏 , and 𝐶 = 2 𝑎𝑏𝑐𝑑 , we obtain 𝑍 (𝑊 ′ ) = 𝑍 (𝑊1 ) + 𝑍 (𝑊2 ) − 𝑍 (𝑊1 )𝑍 (𝑊2 ) (see detailed derivation in (16)). Equality holds if and only if 𝑊1 and 𝑊2 are BECs.

(16)

neous inputs, the polarized subchannel reliabilities are obtained as: 𝑍7 (𝑢1 ) = (2𝜀 − 𝜀 2 ) 3

𝑍7 (𝑢2 |𝑢1 ) = (𝜀 + 𝜀 2 − 𝜀 3 ) (2𝜀 2 − 𝜀 4 )

𝑍7 (𝑢3 |𝑢1 , 𝑢2 ) = 𝜀 3 + 𝜀 4 − 𝜀 7

𝑍7 (𝑢4 |𝑢1 , 𝑢2 , 𝑢3 ) = 𝜀 7

The resulting block error probability is bounded by Î 𝑃 𝐵 (𝑆1 ) ≈ 1 − 4𝑖=1 (1 − 𝑍7 (𝑢𝑖 | . . . )) . • Schedule 𝑆2 (𝑢 2 → 𝑢 3 → 𝑢 4 → 𝑢 1 ): Under the permuted decoding order, the subchannel reliabilities are obtained by 𝑍7 (𝑢2 ) = 𝜀 2 (2 − 𝜀) 2

A PPENDIX B P ROOF FOR CHANNEL - DEPENDENT DECODING

𝑍7 (𝑢3 |𝑢2 ) = 2𝜀 2 − 𝜀 4

𝑍7 (𝑢4 |𝑢2 , 𝑢3 ) = 𝜀 4

SCHEDULING

In this appendix, we provide the rigorous derivation of subchannel reliabilities and block error probability comparisons for the remaining cases 𝑁 = 7 and 𝑁 = 8. These derivations substantiate the claim that the optimal decoding schedule is sensitive to the underlying channel parameter 𝜀 .

𝑍7 (𝑢1 |𝑢2 , 𝑢3 , 𝑢4 ) = 𝜀 3

The resulting block error probability is 𝑃 𝐵 (𝑆2 ) ≈ Î 1 − 𝑖∈ {2,3,4,1} (1 − 𝑍7 (𝑢𝑖 | . . . )) . • Comparison and Threshold Analysis: Proof. The difference in block error probabilities Δ𝑍7 = 𝑃 𝐵 (𝑆1 ) − 𝑃 𝐵 (𝑆2 ) can be expressed as: Δ𝑍7 =

Case 3: Code Length 𝑁 = 7 For 𝑁 = 7, we consider a scenario where the ﬁrst code bit 𝑥1 is punctured. The initial channel reliabilities are deﬁned as 𝑍 (𝑥1 ) = 1 and 𝑍 (𝑥𝑖 ) = 𝜀 for 𝑖 ∈ {2, 3, . . . , 8}. •

Schedule 𝑆1 (𝑢1 → 𝑢2 → 𝑢3 → 𝑢4 ): By applying the generalized recursive relations for heteroge-

− 𝜀 2 (𝜀 − 1) 6 (𝜀 + 1) 2 (𝜀 2 + 1) (𝜀 2 + 𝜀 + 1) · 𝑔(𝜀) = ( Negative) × 𝑔(𝜀)

where 𝑔(𝜀) is a polynomial of the channel parameter. Evaluating at the boundaries, we ﬁnd 𝑔(0) = 6 > 0 and 𝑔(1) = −4 < 0, and 𝑔(𝜀) has a unique real root 𝜀 𝑡 ℎ ≈ 0.920 in the interval (0, 1) .

ZHANG, WANG AND TONG: BEYOND 1→N DECODING: CAPACITY-AWARE RATELESS POLAR CODES FOR IR-HARQ

Consequently, for 𝜀 < 0.920, Δ𝑍7 < 0, rendering 𝑆1 the superior schedule. For 𝜀 > 0.920, the sign ﬂips, and 𝑆2 becomes optimal. Case 4: Code Length 𝑁 = 8 In the non-punctured case (𝑁 = 8), all input channels are homogeneous with 𝑍 (𝑥𝑖 ) = 𝜀 for all 𝑖 . •

Schedule 𝑆1 (𝑢1 → 𝑢2 → 𝑢3 → 𝑢4 ): The standard Arıkan recursion for the selected indices in I yields: 𝑍8 (𝑢1 ) = (2𝜀 − 𝜀 2 ) 4

𝑍8 (𝑢2 |𝑢1 ) = (2𝜀 2 − 𝜀 4 ) 2

𝑍8 (𝑢3 |𝑢1 , 𝑢2 ) = 2𝜀 4 − 𝜀 8

𝑍8 (𝑢4 |𝑢1 , 𝑢2 , 𝑢3 ) = 𝜀 8 •

Schedule 𝑆2 (𝑢2 → 𝑢3 → 𝑢4 → 𝑢1 ): Changing the decoding schedule results in: 𝑍8 (𝑢2 ) = 𝜀 2 (2 − 𝜀) 2

𝑍8 (𝑢3 |𝑢2 ) = 2𝜀 2 − 𝜀 4

𝑍8 (𝑢4 |𝑢2 , 𝑢3 ) = 𝜀 4

𝑍8 (𝑢1 |𝑢2 , 𝑢3 , 𝑢4 ) = 𝜀 4 •

Comparison and Threshold Analysis: Proof. The difference Δ𝑍8 = 𝑃 𝐵 (𝑆1 ) − 𝑃 𝐵 (𝑆2 ) is given by: Δ𝑍8 = 𝜀 2 (𝜀 − 1) 6 (𝜀 + 1) 4 (𝜀 2 + 1) 2 (𝜀 2 − 2𝜀 − 1) · ℎ(𝜀) = ( Negative) × ℎ(𝜀)

where ℎ(𝜀) is the decision polynomial. We observe ℎ(0) = 6 > 0 and ℎ(1) = −1 < 0. The polynomial possesses exactly one real root 𝜀 𝑡 ℎ ≈ 0.965 within the interval (0, 1) . For a reliable channel (𝜀 < 0.965), ℎ(𝜀) is positive, making Δ𝑍8 < 0. Thus, the sequential schedule 𝑆1 is optimal. As the channel degrades toward the threshold 𝜀 𝑡 ℎ ≈ 0.965, the optimal decoding order switches to 𝑆2 . ACKNOWLEDGMENT The authors would like to express their sincere gratitude to Prof. Erdal Arıkan for his valuable suggestions and constructive feedback during the preparation of this manuscript. His insights were instrumental in improving the quality and clarity of this work.

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