Circular Buffer Rate Matching for Polar Codes

• Circular buffer rate matching (CB-RM) is a unified framework that integrates puncturing, shortening, and repetition for constructing rate-compatible polar codes adaptable to diverse channel conditions.
• It employs a two-stage polarization process and a progressive puncturing algorithm based on binary domination, ensuring capacity-achieving performance with nested code families.
• CB-RM supports BICM and higher-order modulations, delivering low-complexity encoding/decoding and measurable performance gains in practical wireless and HARQ systems.

Circular buffer rate matching (CB-RM) is an advanced framework for constructing rate-compatible polar codes, achieving flexible code rates for diverse channel conditions and HARQ applications by integrating puncturing, shortening, and repetition within a unified operational and mathematical paradigm. The approach enables seamless adaptation to single- or multi-transmission channels and provides capacity-achieving, low-complexity solutions for both binary-input and higher-order modulated communication systems (El-Khamy et al., 2017, Jang et al., 2019).

1. Mathematical Foundations of the Circular-Buffer Operator

At the core of CB-RM, the mother polar codeword $x_1^N$ (with $N=2^n$) is arranged as a $2^q\times2^p$ matrix $X(i,j)$, where $p+q=n$ and $X(i,j)=x_{(i-1)2^p+j}$ for $1\leq i\leq2^q, 1\leq j\leq2^p$. The central operation is the rate-matching map,

$$\mathcal{R}(x_1^N;L) = (\hat{x}_1, \hat{x}_2, \ldots, \hat{x}_L),$$

with

$$\hat{x}_t = X(1 + ((t-1) \bmod 2^q),\; \pi(\lfloor (t-1)/2^q \rfloor + 1)),$$

where the permutation $\pi$ (derived from the progressive puncturing algorithm or a $2^n$-posequence) determines the column-interleaving order. For all $L$, exactly the worst $\lceil N-L\rceil$ columns in $\pi$ order are excluded (for $L<N$, i.e., puncturing), while $L>N$ (repetition) simply cycles through the codeword with wraparound.

Unified CB-RM, as developed via binary domination, produces the codeword $c_k = x'_{k \bmod N}$ for $k = 0 \ldots M-1$ after applying an interleaver $\pi$ to the encoder output $\mathbf{x}$, ensuring one uniform buffer extraction mechanism for all modes (puncturing, shortening, repetition) (Jang et al., 2019).

2. Two-Stage Polarization Structure

The encoder is represented as $$F_2^{\otimes n} = (F_2^{\otimes q}) \otimes (F_2^{\otimes p})$$ with a subsequent bit-reversal operation. This induces a two-stage polarization process:

• Stage 1 applies $2^q$ parallel base codes of length $2^p$, polarizing the input into $N'$ bit-channels $\{\tilde{W}_{2^p}^{(i)}\}$.
• Stage 2 applies $2^p$ parallel transforms to these bit-channels.

In the presence of puncturing, exactly $m$ out of $2^p$ base code channels become zero-capacity, reducing the achievable rate to $(1 - \frac{m}{2^p})I(W)$ in the asymptotic regime. This nested polarization ensures that all derived codes for different rates share a common structure, enabling fully nested, rate-compatible code families (El-Khamy et al., 2017).

3. Progressive Puncturing Algorithm and Binary Domination

The progressive puncturing algorithm (PPA) selects puncturing patterns iteratively on the base code of length $N'=2^p$. At each step, it identifies the position $j^\ast$ that, when punctured, minimally increases the estimated SC-decoding error probability for information bit-channels. The resulting permutation $\pi$ orders columns (or bits) by vulnerability under puncturing, giving rise to a nested sequence of puncturing sets.

Binary domination provides a formal equivalence and ordering: for indices $i,j$ with binary expansions, $i \preceq j$ (read "$j$ dominates $i$") if $i_t\leq j_t$ for all binary digits. This yields down-sets in the partial order, fundamental to characterizing incapable (for puncturing) or fixed (for shortening) bits:

• If an input $u_j$ becomes incapable, so do all $u_i$ with $i \prec j$.
• For shortening, fixing $u_j=0$ forces all outputs $x_k$ with $j \preceq k$ to zero (Jang et al., 2019).

Posequences (topological orderings under $\preceq$) are used not only for defining puncturing order, but also for aligning the design of unified buffer-extraction for both puncturing and shortening.

4. Unified Buffer Mapping: Algorithms and Equivalence

The circular buffer rate-matching interleaver is constructed from the chosen posequence $P$, with $\pi(i)=p_i$ for the $i$th entry. The codeword is then extracted as $c_k = x'_{k\bmod N}$ for $k=0,\ldots,M-1$. This methodology unifies:

• Puncturing: Exclude the first $J$ entries of $\pi$ for incapable bits.
• Shortening: Use the last $J$ entries for bits to be fixed, with bitwise complement yielding the equivalent incapable set.

The equivalence of "high-index" and "low-index" bit puncturing is formally established: puncturing either the first or last $J$ entries of a posequence yields (via binary complement) the same incapable pattern, guaranteeing identical code properties. This unification eliminates the need for separate hardware logic or buffer management between puncturing and shortening scenarios (Jang et al., 2019).

Example Table: Posequence and Buffer Extraction

$N$	Posequence $P$	Operation	Indices Punctured/Shortened	Codeword Extraction
8	(0,1,2,4,3,5,6,7)	Puncturing	$\{0,1\}$	$c_k = x'_{k},\ k=0..5$
8	(0,1,2,4,3,5,6,7)	Shortening	$\{6,7\}$	$c_k = x'_{k},\ k=0..5$

5. Bit-Mapping to BICM and Capacity Arguments

CB-RM is naturally extended to bit-interleaved coded modulation (BICM) and higher-order modulations (e.g., $M$-QAM with Gray labeling). Here, $\ell = \log_2 M$ binary subchannels of differing reliability are grouped by the interleaver into $2^p$ compound subchannels. Of these, $2^p - m$ active columns are uniformly mapped; punctured columns serve as zero-capacity channels.

The capacity result, invoking compound polarization, states: $$\lim_{N\to\infty}\frac{1}{N}|\text{good channels}| = \left(1 - \frac{m}{2^p}\right)\frac{1}{\ell}\sum_{j=1}^{\ell}I(W_j),$$ where the rightmost factor is precisely the symmetric BICM capacity. Thus, the CB-RM framework remains capacity-achieving in BICM environments under SC decoding (El-Khamy et al., 2017).

6. Performance, Complexity, and Implementation

Empirical performance and complexity data indicate:

• Finite-length CBRM-polar codes exhibit BER close to exhaustive-search punctured codes and outperform LTE-Turbo codes (by 0.2–0.4 dB in mid-length settings) and SC-QC-LDPC codes (by >1 dB at high rates).
• HARQ implementations (e.g., IR–HARQ with 16-QAM) gain ≈3 dB over Chase–Combining.
• Encoding and SC-decoding complexity remains $O(N\log N)$, with no additional large table requirements except the puncturing/posequence $\pi$.
• Throughput benefits are observed with unified CB-RM logic due to elimination of per-mode branching and simplified buffer-extraction, yielding modest but measurable increases in practical settings (El-Khamy et al., 2017, Jang et al., 2019).

7. Significance and Practical Impact

Circular-buffer rate matching provides a unified, provably optimal foundation for constructing versatile polar code families. Its efficient hardware/software compatibility, integrated BICM support, and guaranteed correct incapable/fixed bit patterns by binary domination advance both theoretical understanding and implementation practicality for modern wireless and coded-modulation standards (El-Khamy et al., 2017, Jang et al., 2019). The framework's flexibility, capacity-achieving guarantees, and simplicity of rate adaptation underpin its adoption in systems requiring robust, rate-flexible channel coding.