Computer Science – Information Theory
Scientific paper
2012-03-15
Computer Science
Information Theory
30 pages, 11 figures, submitted to IEEE Transactions on Information Theory
Scientific paper
It is well known that dense lattice packings can be obtained via Construction $A$ from \emph{binary} linear codes. In this paper, we propose an extension of Construction $A$ called Construction $A^\prime$ to obtain Barnes-Wall lattices from linear codes over finite rings. To obtain the Barnes-Wall lattice $BW_{2^{m}}$ in $\mathbb{C}^{2^{m}}$ for any $m \geq 1$, we identify a linear code $\mathcal{C}_{2^m}$ over a polynomial ring and then embed the linear code to a lattice constellation $\mathcal{L}_{2^{m}}$ such that $BW_{2^{m}}$ can be obtained as $BW_{2^{m}} = (1+i)^{m}\mathbb{Z}[i]^{2^{m}} \oplus \mathcal{L}_{2^{m}}$ where $i = \sqrt{-1}$. We also show that $\mathcal{L}_{2^{m}}$ has the cubic shaping property when $m$ is even. We highlight that Construction $A^\prime$ provides a convenient technique for bit-labelling Barnes-Wall lattice constellations. We also employ the lattice constellation $\mathcal{L}_{2^{m}}$ as a coded modulation scheme for AWGN channels. To encode the code, we use Construction $A^{\prime}$, and to decode the code we use the infinite Barnes-Wall lattice decoder (IBWD) proposed by Micciancio and Nicolosi. First, we study the error performance of IBWD in decoding the infinite lattice and then propose a variant of it called the Barnes-Wall lattice constellation decoder (BWCD) to decode the lattice constellation. Simulation results on the bit error rate of BWCD are also presented. This work is a step towards constructing polar lattice codes.
Belfiore Jean-Claude
Harshan J.
Viterbo Emanuele
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