Chapter Summary

Chapter Summary

Key Points

• 1.

  Lattice codes provide structured alternatives to random Gaussian codebooks. A lattice code is formed by intersecting a lattice $\Lambda \subset \mathbb{R}^n$ with a shaping region. As the dimension $n$ grows, the normalized second moment $G(\Lambda) \to 1/(2\pi e)$, and lattice codes achieve AWGN capacity (Erez-Zamir, 2004).

• 2.

  Three families of practical codes approach the Shannon limit: turbo codes (parallel concatenation + iterative decoding, 1993), LDPC codes (sparse parity-check + belief propagation, 1962/1996), and polar codes (channel polarization + successive cancellation, 2009). Only polar codes have a rigorous proof of capacity-achieving performance with explicit construction.

• 3.

  Channel polarization transforms $N$ copies of a binary-input channel $W$ into $N$ virtual channels that polarize: fraction $I(W)$ become nearly perfect and fraction $1 - I(W)$ become nearly useless. Information bits go on the good channels; frozen bits on the bad ones. Encoding and decoding both run in $O(N\log N)$.

• 4.

  The gap to capacity decomposes into coding gain ($< 0.5$ dB with modern codes) and shaping gain (up to $\pi e/6 \approx 1.53$ dB). Probabilistic shaping uses a Maxwell-Boltzmann distribution over constellation points to approximate the Gaussian input, closing the shaping gap independently of the FEC code.

• 5.

  5G NR deploys LDPC codes for data channels and polar codes for control channels, reflecting their complementary strengths: LDPC for high-throughput long blocks, polar (with CA-SCL decoding) for reliable short blocks.