4.10.3 Polar Codes

Polar codes represent a third major milestone in capacity-approaching error-control coding, following Turbo and LDPC codes. Proposed by E. Arıkan in 2008, they are the first provably capacity-achieving block codes for binary-input, memoryless channels with low encoding and decoding complexity

The central idea of channel polarization is to transform a set of identical channels into two extreme groups—completely reliable and completely unreliable—by recursively combining and splitting them through a linear transform. Information bits are transmitted over the reliable channels, while fixed “frozen” bits occupy the unreliable ones. As the block length increases, the proportion of reliable channels approaches the channel capacity, allowing polar codes to achieve that limit asymptotically.

Polar codes employ a recursive construction based on the Kronecker power of a simple 2×2 kernel. Encoding consists of a series of bit-wise XOR and permutation stages, which can be implemented efficiently in O(NlogN) time. Decoding relies on successive-cancellation (SC) or its improved successive-cancellation-list (SCL) algorithm, in which each bit decision conditions the decoding of subsequent bits. Although SC decoding performs poorly for short blocks, SCL decoding with a CRC approaches LDPC-like performance at moderate lengths while maintaining deterministic complexity.

Unlike Turbo or LDPC codes, which rely on iterative message passing between constituent decoders, polar codes are monolithic but structured recursively. They may thus be regarded as a non-iterative compound code: multiple “polarized” sub-channels interact deterministically through the transformation rather than stochastically through iterations. This property yields predictable scaling laws and guarantees convergence to capacity without the need for random interleaving or sparse-graph optimization.

Polar codes are particularly attractive for control and short-packet links where latency and power efficiency are critical. They offer:

Low-complexity encoding and decoding suitable for onboard or user-terminal processors.
Excellent performance for short and moderate block lengths, matching the bursty framing of return-link and telemetry channels.
Deterministic latency, beneficial for command, tracking, and synchronization signals.

In satellite systems, LDPC codes remain the preferred choice for high-throughput forward links (e.g., DVB-S2/S2X and CCSDS 131.0-B-4), while polar codes are gaining attention for return-link and low-latency control channels. Their adoption as the 5G NR control-channel code by 3GPP Release 17 has further strengthened their role in emerging non-terrestrial networks (NTN) that integrate terrestrial, satellite, and high-altitude-platform segments.

Polar codes have also been proposed for next-generation CCSDS telemetry and Earth-observation missions as short-frame alternatives to BCH and convolutional codes, and as candidates for hybrid Turbo–polar concatenations that combine the iterative gains of Turbo decoding with the structural regularity of polarization.

Polar codes approach the Shannon limit with increasing block length and, under SCL decoding, can outperform Turbo codes for short to medium lengths. They achieve excellent BER performance with significantly lower computational complexity than iterative schemes and provide native support for rate matching through puncturing and shortening. Their structured design lends itself to hardware parallelization and scalability across different link configurations.