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Design of Polar Codes in 5G New Radio

Valerio Bioglio, Carlo Condo, Ingmar Land

arXiv:1804.04389v3cs.IT

TL;DR

5G polar-code design must support flexible lengths and rates while keeping encoding and decoding practical. The paper presents a tutorial of the standardized encoding chain and its associated reliability, rate-matching, and decoding techniques. The resulting framework targets low description and encoding complexity across eMBB control-channel configurations.

  • Problem

    5G requires polar codes across diverse lengths, rates, and channel conditions, while on-the-fly reliability design can impose excessive coding latency.

  • Method

    The paper gives a self-contained tutorial of 5G-compliant polar encoding, including concatenation, interleaving, subchannel allocation, rate matching, and decoding considerations.

  • Results

    The standardized framework covers wide code-length and code-rate ranges while targeting low description complexity, low encoding complexity, and feasible decoder complexity.

  • Takeaways & Limitations

    The detailed description can serve as a reference for understanding, implementing, simulating, and further developing 5G polar codes.

Abstract

from arXiv · show

Polar codes have attracted the attention of academia and industry alike in the past decade, such that the 5$^\text{th}$ generation wireless systems (5G) standardization process of the 3$^\text{th}$ generation partnership project (3GPP) chose polar codes as a channel coding scheme. In this tutorial, we provide a description of the encoding process of polar codes adopted by the 5G standard. We illustrate the struggles of designing a family of polar codes able to satisfy the demands of 5G systems, with particular attention to rate flexibility and low decoding latency. The result of these efforts is an elaborate framework that applies novel coding techniques to provide a solid channel code for NR requirements.

I. INTRODUCTION

Polar codes were adopted for 5G control information, but supporting diverse lengths, rates, and channel conditions requires low-complexity, flexible designs. This tutorial explains the standardized encoding chain and its implementation-oriented structure.

  • Polar codes were adopted by 3GPP for uplink and downlink control information in 5G eMBB.
  • 5G requires codes across varied lengths, rates, and channel conditions, making separate reliability vectors for every combination impractical.
  • The standardized encoding framework combines techniques from competing design agendas, yielding acceptable performance rather than a single pinnacle solution.
  • The paper provides a self-contained tutorial of 5G encoding, covering concatenation, interleaving, subchannel allocation, and rate matching.
  • The tutorial is organized around polar-code fundamentals, advanced design features, encoding, decoding considerations, and conclusions.

B. Code design

Polar-code design builds larger transforms from basic polarization kernels and selects reliable synthetic channels for information transmission. The recursive structure supports efficient encoding while 5G rate matching addresses non-power-of-two codeword lengths.

  • Concatenating basic polarization kernels forms G_N = G_2^⊗n and accelerates polarization while limiting encoding and decoding complexity.
  • An (N, K) polar code selects the K most reliable synthetic channels for information bits and freezes the remaining N − K channels.
  • The auxiliary input vector assigns zeros to frozen positions and places information bits in the remaining positions before codeword generation.
  • The recursive transform enables parallel matrix operations, reducing encoding complexity to O(log2(N)) and organizing decoding into log2(N) steps.

D. Decoding

5G decoding follows the standardized encoding structure in reverse, using polar-code tree algorithms and rate-matching-dependent padding. Frozen-set construction and list-based decoding address reliability, latency, and error-correction requirements.

  • Successive cancellation decoding traverses a binary tree, estimates leaf bits, and always decodes frozen bits as zero.
  • For a BEC, SC decoding propagates soft values in {0, 1, e} using f and g update rules, then combines hard decisions through the tree.
  • On-the-fly reliability estimation can be computationally costly or too slow for 5G latency requirements, motivating standardized frozen-set design.
  • The 5G standard uses a universal sequence of 1024 reliability-ordered bit-channel indices to derive frozen sets for codes of length at most 1024.
  • SCL improves SC error correction at moderate lengths by retaining multiple candidate paths, while larger list sizes increase implementation complexity.

C. Rate matching

5G rate matching adapts power-of-two mother polar codes to arbitrary codeword lengths using puncturing, shortening, extension, and repetition. The design balances reliability, decoding latency, implementation complexity, and higher-order modulation requirements.

  • Rate matching: 5G must match fixed information length A to a required codeword length E, despite mother polar codes being restricted to N=2^n.Puncturing, shortening, and extending solve this length-matching problem.
  • Puncturing and shortening: Puncturing treats untransmitted bits as decoder erasures, whereas shortening fixes them to known values, typically zero.Puncturing creates incapable bit-channels, while shortening creates overcapable channels but requires matching-pattern bits to depend only on frozen bits.
  • Design trade-offs: 3GPP favors frozen-set-based matching patterns because on-the-fly reliability recalculation increases latency and joint optimization has excessive complexity.This choice reduces code-design complexity at the cost of increased BLER.
  • Extension: When E is slightly above a power of two, extending a smaller mother code by retransmitting code bits can reduce decoding latency and improve error-correction performance.A smaller mother-code decoder can combine the repeated likelihoods.
  • Polar-coded modulation: BIPCM provides a low-complexity alternative to polar-coded modulation for high-order modulation and improves diversity gain without increasing code complexity.Careful mapping of coded bits to modulation symbols can outperform conventional random interleaving.

IV. POLAR CODE ENCODING IN 5G

The 5G polar encoding chain transforms information bits into transmitted code bits through optional segmentation, CRC processing, interleaving, polar encoding, sub-block interleaving, rate matching, and channel interleaving. Its parameters and operations vary across uplink and downlink channels to support their admitted message and payload ranges.

  • Scope: 5G uses polar codes for UCI on PUCCH/PUSCH and for DCI on PDCCH and PBCH payloads.The encoding framework therefore supports distinct uplink and downlink channels.
  • Input processing: The chain optionally segments the message, appends an L-bit CRC to each segment, interleaves the resulting K=A′+L bits, and places them into a mother polar code.Segmentation allows separate encoding of two parts when required by the code parameters.
  • Output processing: After encoding, a sub-block interleaver creates y, rate matching converts N bits to E bits through puncturing, shortening, or repetition, and channel interleaving produces f for modulation.Concatenation may follow before transmission as g.
  • Channel-specific parameters: Uplink parameters allow 12≤A≤1706 and 18≤E≤8192, while downlink PDCCH limits A to 140 and PBCH uses A=32 and E=864.The payload length G may reach 16384 and can trigger segmentation in the uplink.
  • Mother-code selection: The mother-code exponent is bounded by n_min=5, n_max=9 for downlink control, and n_max=10 for uplink, with additional bounds determined by rate and matching.If N>E, puncturing or shortening is selected by code rate; if N<E, repetition is applied.

C. CRC encoding

CRC encoding appends channel-specific parity checks to each message segment before polar encoding. The standard selects polynomial, initialization, and masking rules according to the payload or control channel.

  • CRC construction: An L-bit CRC is appended to the A′ message bits, producing a vector c of A′+L bits.The CRC output is subsequently used in the polar encoding chain.
  • Channel-specific polynomials: The 24-bit polynomial g24(x) is used for PBCH payloads and PDCCH DCIs, while g6(x) and g11(x) serve UCIs with 12≤A≤19 and A≥20, respectively.The longer CRC supports payloads and DCIs requiring more assistant bits for early termination.
  • Initialization: The CRC shift register starts at all zeros for UCIs and PBCH payloads, but at all ones for DCIs.This initialization rule is part of the channel-specific CRC procedure.
  • DCI masking: For DCIs, the last 16 CRC parity bits are masked using the radio network temporary identifier.The masking is expressed through XOR with the RNTI-derived sequence.

D. Input bits Interleaver

5G distributes CRC-augmented bits across the polar information set, computes assistant and frozen bits, and uses a universal reliability sequence to construct the mother-code input. The procedure adapts freezing to rate matching and protects against incapable channels.

  • Input-bit interleaving: The input-bit interleaver permutes CRC-encoded bits so each CRC remainder bit follows the information bits involved in its parity check.This distributed CRC arrangement supports early termination and is enabled for PBCH and PDCCH but bypassed for PUCCH and PUSCH.
  • Interleaver construction: The interleaver pads inputs to K_max^IL=164 bits and derives the active permutation from the maximum sequence Π_max^IL.The final interleaved vector is c′={c_Π(0),…,c_Π(K−1)}.
  • Information and assistant bits: The N-bit vector u contains interleaved information and CRC bits, n_PC parity-check bits, and frozen zeros, making the mother code dimension K′=K+n_PC.Frozen-set selection depends on N, E, and the selected rate-matching scheme.
  • Reliability sequence: The universal reliability sequence orders all indices below 1024 from least to most reliable, and indices below N are extracted in relative order for a mother code.For example, N=512 selects the red through azure ranges, while N=256 selects the red through green ranges.
  • Rate-matching interaction: Rate matching identifies frozen indices from the untransmitted codeword positions, with puncturing and shortening using different ends of the rate-matching pattern.When puncturing applies, additional indices are frozen to prevent information positions from becoming incapable.
  • Frozen-set design: The frozen set is formed from pre-freezing, extra freezing, and reliability freezing, then the information set is its complement and frozen input positions are set to zero.Pre-freezing handles untransmitted bits, extra freezing addresses puncturing conditions, and reliability freezing fills the remaining positions.

2) Subchannel allocation:

The framework allocates message and parity-check bits across polar-code subchannels using complementary selection strategies based on reliability and generator-matrix row weight.

  • Parity-check subchannels are selected using either generator-matrix row weight or subchannel reliability.
  • Initially, n_lr^PC parity-check indices are assigned to the least reliable subchannels among the available indices.
  • The remaining n_wm^PC index is selected from the K most reliable available indices by choosing the minimum-weight transformation-matrix row.
  • When row weights tie, the subchannel with the highest reliability is selected; row weight is w(g_i) = 2^o_i.
  • The K message bits are stored in the K designated indices of the encoded vector.

I \QN

After the parity-check subchannels are allocated, the remaining parity-check indices are calculated.

  • After parity-check subchannels have been allocated, the values of the remaining n_PC indices are calculated.
  • The allocation step precedes calculation of the remaining parity-check values.
  • The passage distinguishes allocated parity-check subchannels from the remaining indices whose values are subsequently calculated.

3) PC bit calculation:

After encoding, 5G polar-code bits undergo sub-block interleaving and circular-buffer rate matching, supporting puncturing, shortening, or repetition according to code length and rate.

  • 3) PC bit calculation:: Parity-check bits are computed with a length-5 cyclic shift register initialized to zero, using preceding message bits modulo 5.
  • 3) PC bit calculation:: The polar transform encodes the vector through multiplication over F2, with complexity O(N log2(N)).
  • 3) PC bit calculation:: The N encoded bits are interleaved by dividing them into 32 blocks of length B = N/32 and permuting those blocks.
  • 3) PC bit calculation:: Rate matching uses a circular buffer with puncturing when E ≤ N and R ≤ 7/16, shortening when E ≤ N and R > 7/16, and repetition when E > N.

I. Channel interleaver

The optional channel interleaver uses a triangular structure to reorder rate-matched bits before modulation, improving high-order-modulation coding performance while supporting parallel and flexible implementation.

  • I. Channel interleaver: The channel interleaver is optional and is activated for PUCCH and PUSCH UCIs but bypassed for PBCH payloads and PDCCH DCIs.
  • I. Channel interleaver: It uses an isosceles triangular structure of length T bits, with encoded bits written into rows.
  • I. Channel interleaver: The interleaved vector is formed by reading the triangular structure column-wise while skipping NULL entries.
  • I. Channel interleaver: The triangular design was proposed for 5G because of high parallelism, maximum contention-free behavior, and flexibility.

J. Block concatenation

The 5G polar-code chain combines segmentation, rate matching, interleaving, and subchannel processing, with corresponding implications for decoding and performance. The resulting structure supports practical decoding choices and exposes trade-offs between error rate, false-alarm mitigation, and termination efficiency.

  • Block concatenation: When segmentation is active, two length-E codewords are appended to form a length-G block, with a zero appended when G = 2E + 1.
  • Decoding chain: Decoding can invert the encoding chain, including uplink block segmentation and deinterleaving before padding to the mother polar-code length.The padding pattern depends on the rate-matching method.
  • Decoding chain: SCL decoding processes assistant bits, then may deinterleave and concatenate bits according to code and channel parameters before passing them upward.SCL is described as the de-facto standard decoder for these polar codes.
  • Assistant bits: CRC and PC assistant bits support early termination, false-alarm reduction, or BLER improvement, with their roles depending on how they are incorporated.Distributed CRC bits target early termination and false-alarm mitigation, while PC bits target error correction.
  • Assistant bits: Failure handling in list decoding affects BLER: retaining failing paths preserves list size and simplifies implementation, while deactivation is an alternative.Early termination continues when at least one active path passes the check.
  • Performance considerations: Figure 13 evaluates SNR for BLER 10^-3 under BPSK-AWGN with list size 8; PC bits improve PUCCH performance, whereas PDCCH trades BLER for FAR mitigation and earlier termination.The PDCCH degradation is attributed to its larger CRC, while maintaining failing paths in the SCL list.
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