Chapter 17: LAST Codes — Lattice Space-Time Codes

Learning Objectives

• State the LAST-code construction: a fine lattice $\Lambda_c \subset \mathbb{R}^{n_t T}$ shaped by a coarse lattice $\Lambda_s$, with codewords $\mathbf{X} = \mathbf{G} \mathbf{u} \bmod \Lambda_s$ and common random dithering, explain how LAST relocates the $\Lambda/\Lambda$-coded symbols onto the MIMO channel, and identify the random-LAST and structured-LAST variants
• Derive the MMSE-GDFE (Minimum-Mean-Square-Error Generalised Decision-Feedback Equaliser) from the QR decomposition of the augmented matrix $[\mathbf{H}^{T}, \sqrt{\alpha}\mathbf{I}]^T$, identify the MMSE coefficient $\alpha = 1/\text{SNR}$, and explain how this transforms the MIMO channel into $n_t$ triangular layers with aggregate SNR equal to the full MIMO SNR
• State and prove the El Gamal-Caire-Damen 2004 theorem: LAST codes with MMSE-GDFE lattice decoding achieve the Zheng-Tse diversity-multiplexing tradeoff $d^*(r) = (n_t - r)(n_r - r)$ for every $(n_t, n_r)$ and every $r \in [0, \min(n_t, n_r)]$, and recognise that this proof is the information-theoretic counterpart of the algebraic CDA-NVD proof of Ch. 13
• Apply the Kumar-Caire 2008 construction: use $E_8$ or the Leech lattice $\Lambda_{24}$ as the inner lattice of a LAST code, preserve DMT optimality, and gain 3-6 dB of coding-gain improvement over random LAST at moderate SNR; recognise the dimension-matching constraint $n_t T \in \{8, 16, 24, \ldots\}$
• Compare the five canonical MIMO decoders — brute-force ML, sphere decoder, MMSE-GDFE lattice, K-best, ZF-slicing — in complexity, performance, and DMT-achievability; explain why MMSE-GDFE + lattice decoding is the sweet spot for LAST codes and why 5G NR nonetheless uses linear MMSE with BICM outer codes
• Connect LAST to the broader landscape: backward to CDA codes (Ch. 13, algebraic DMT), to Erez-Zamir lattice AWGN codes (Ch. 16, lattice AWGN capacity), and forward to compute-and-forward (Ch. 18, lattice network coding)