Polar Orbit Decoding (POD)

• Polar Orbit Decoding is a universal decoding framework that embeds any binary linear block code into a polar subcode using dynamic-frozen constraints.
• It leverages automorphism groups to generate permutation orbits, enabling parallel soft-decoding streams that approach near-maximum-likelihood performance.
• The framework reduces hardware complexity by unifying diverse BLBCs under a single polar decoder architecture, balancing latency and error-rate tradeoffs.

Polar Orbit Decoding (POD) is a universal, parallel soft-decoding framework for binary linear block codes (BLBCs), designed to enable near-maximum-likelihood (ML) performance with low latency and universal hardware compatibility. By exploiting code automorphisms to generate distinct permutation orbits and embedding any BLBC into the polar code framework with dynamic-frozen constraints, POD unifies the decoding process for diverse code families within a single decoder architecture (Li et al., 16 Jan 2026).

1. Universal Representation of Binary Linear Block Codes

A binary linear block code $C$ over $\mathbb{F}_2$ of length $n$ and dimension $k$ is described by a $k \times n$ generator matrix $G$:

$C = \{c = mG \mid m \in \mathbb{F}_2^{k} \}.$

Traditional code families (BCH, Golay, Reed–Muller, LDPC) offer tradeoffs between minimum distance, complexity, and asymptotics. No single family is optimal across all metrics; consequently, standards such as 5G implement multi-code architectures, demanding multiple decoder types per device and multiplying hardware complexity.

Universal polar decoding circumvents this by embedding any BLBC into a polar code—specifically, as a polar subcode with “dynamic frozen” bits. This process uses:

• A length-$n$ polar generator $G_n = B_n F^{\otimes m}$, with $n=2^m$, $\mathbb{F}_2$0, and $\mathbb{F}_2$1 as the bit-reversal permutation.
• A permutation matrix $\mathbb{F}_2$2.
• The code is mapped via $\mathbb{F}_2$3, then row-reduced to form an upper-trapezoidal matrix $\mathbb{F}_2$4 of dynamic frozen constraints:

$\mathbb{F}_2$5

This expresses the original BLBC as a polar subcode; constraints in $\mathbb{F}_2$6 are enforced during polar-style decoding (successive-cancellation [SC], or SC-list [SCL]).

Significance: Universal embedding permits one hardware decoder to process any BLBC, conditional on efficient and flexible polar decoding.

2. Automorphism Groups and Permutation Orbits

The automorphism group $\mathbb{F}_2$7 of a code $\mathbb{F}_2$8 with generator $\mathbb{F}_2$9 is:

$n$0

This comprises all coordinate permutations mapping the code onto itself—equivalently, $n$1 if permuting $n$2’s columns via $n$3 yields a matrix with the same row space.

Invariant Dynamic Frozen Sets: For any polar embedding permutation $n$4, every $n$5 gives rise to $n$6 with the same dynamic frozen matrix $n$7. That is, for all $n$8,

$n$9

produces identical polar-constraint matrices after RREF, so the subcode constraints are invariant under automorphism-induced permutations.

Automorphism Orbits: Fixing $k$0, the associated orbit is

$k$1

Each $k$2 defines a distinct embedding trajectory yet preserves the dynamic frozen constraints. Decoding the received word $k$3 under every element in the orbit provides multiple diverse soft-decoding perspectives, all aligned to the same polar subcode.

This invariance eliminates the need for per-permutation adaptation of the frozen set logic, streamlining hardware implementation.

3. Parallel Decoding Algorithm and Workflow

POD operates by decoding in parallel over $k$4 elements of the automorphism orbit $k$5. The procedural workflow is:

1. Orbit Enumeration: Precompute $k$6 distinct automorphisms $k$7 by subgroup enumeration.
2. Parallel Decoding:
   • For each $k$8, permute the channel log-likelihood ratio (LLR) vector $k$9 by $k \times n$0, producing $k \times n$1.
   • Input $k \times n$2 into a polar-style decoder (SC, SCL of list-size $k \times n$3, or related dynamic-frozen bit engine).
   • Collect candidate paths $k \times n$4 for $k \times n$5.
3. Candidate Aggregation: Gather all $k \times n$6 candidate paths, invert each permutation, and apply a combiner (minimum path metric or BLBC parity check).
4. Output Selection: Select the valid candidate with optimal path metric.

Pseudocode Outline:

$n=2^m$9 Each PolarListDecode maintains SC path metrics and prunes to the top $k \times n$7 survivors per bit index. The frozen-set logic ($k \times n$8) remains identical for all decoders.

The effective list size is $k \times n$9. Varying $G$0 yields control over the tradeoff between latency and performance.

4. Automorphism Orbit Construction: BSGS and Schreier–Sims Methodology

Automorphism groups for BLBCs can be vast (e.g., extended BCH: $G$1; Golay $G$2: $G$3), necessitating efficient orbit enumeration. POD adopts a base and strong generating set (BSGS) representation via the Schreier–Sims algorithm:

• Base Selection: Choose a sequence of points $G$4 in $G$5.
• Transversal and Generator Refinement: Iteratively refine transversal sets and Schreier generators, maintaining stabilizer chains $G$6. Schreier’s Lemma extracts strong generators for $G$7.
• Complexity: BSGS construction is $G$8 for a generating set $G$9 of size $C = \{c = mG \mid m \in \mathbb{F}_2^{k} \}.$0; for algebraic codes, this remains polynomial in $C = \{c = mG \mid m \in \mathbb{F}_2^{k} \}.$1.

With BSGS precomputed, enumeration or random sampling of $C = \{c = mG \mid m \in \mathbb{F}_2^{k} \}.$2 orbit elements costs $C = \{c = mG \mid m \in \mathbb{F}_2^{k} \}.$3 backend operations, negligible for hardware accelerators.

5. Core Formulas and Computational Tradeoffs

The polar transform and codeword mapping utilize:

• Polar Generator:

$C = \{c = mG \mid m \in \mathbb{F}_2^{k} \}.$4

• Message and Constraint Application:

$C = \{c = mG \mid m \in \mathbb{F}_2^{k} \}.$5

Within SCL decoding, at each bit index $C = \{c = mG \mid m \in \mathbb{F}_2^{k} \}.$6:

• Log-likelihood ratio update:

$C = \{c = mG \mid m \in \mathbb{F}_2^{k} \}.$7

• Path metric update:

$C = \{c = mG \mid m \in \mathbb{F}_2^{k} \}.$8

Latency, list-size, and error-rate are characterized by:

• SC latency: $C = \{c = mG \mid m \in \mathbb{F}_2^{k} \}.$9
• SCL sorting step: $n$0
• SCL decoding: $n$1
• POD with $n$2 parallel SC streams: per-stream latency $n$3, plus $n$4 for combining. Effective list size $n$5 yields performance between $n$6 and $n$7.

6. Performance and Simulation Analysis

Simulations employ additive white Gaussian noise (AWGN) channels, binary phase-shift keying (BPSK), and BLBC-native parity checks for candidate screening. Results span several BLBCs:

Code	POD Configuration	ML Performance Approximation	Latency Profile
(16,7) eBCH	POD$n$8–SC	Matches SCL$n$9/ML at high SNR	$G_n = B_n F^{\otimes m}$0
(64,16) eBCH	POD$G_n = B_n F^{\otimes m}$1–SC	Collides with SCL$G_n = B_n F^{\otimes m}$2/ML	SC-per-stream
(64,36) eBCH	POD$G_n = B_n F^{\otimes m}$3–SC	Halves error versus SC, near SCL$G_n = B_n F^{\otimes m}$4	SC-per-stream
(24,12) Golay	POD$G_n = B_n F^{\otimes m}$5–SC	$G_n = B_n F^{\otimes m}$61 dB gain over SC, approaches ML	SC-per-stream

Increasing $G_n = B_n F^{\otimes m}$7 interpolates error performance between SC and ML with commensurate decreases in decoding latency. Using SCL as the internal decoder (e.g., POD$G_n = B_n F^{\otimes m}$8–SCL$G_n = B_n F^{\otimes m}$9) enables further error-rate improvements (e.g., outperforming SCL$n=2^m$0).

This suggests that parallelism via orbit decoding provides a scalable, hardware-efficient route to near-ML performance with controllable tradeoffs.

7. Hardware Implementation and Complexity Considerations

POD hardware instantiates a single polar decoder core ($n=2^m$1 times in parallel). Each core maintains identical frozen-set logic (matrix $n=2^m$2); no duplicated constraint computation occurs. BSGS representation for the automorphism group $n=2^m$3 is stored once, with each $n=2^m$4 retrieved via $n=2^m$5 pointer-chasing operations.

The final candidate combine step is a modest comparison tree with BLBC code validity checks. The hardware complexity thus scales with the parallel stream count $n=2^m$6, trading increased decoder cores for reduced wall-clock latency. The universal decoder core is portable across BLBCs given appropriate $n=2^m$7 and $n=2^m$8 parameters.

A plausible implication is that universal, parallel polar decoding hardware based on POD may standardize and consolidate BLBC decoder design in future communication systems, reducing the need for heterogeneous multi-code decoder architectures.

Summary

Polar Orbit Decoding employs the automorphism group of any BLBC to produce multiple equivalent polar subcode embeddings, enabling universal, parallel soft decoding. These orbit-induced embeddings facilitate parallel SC or SCL decoding, yielding near-ML performance at SC-level latency and with hardware universality across BLBC classes (Li et al., 16 Jan 2026).