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- W2963027968 abstract "Encoding and indexing of lattice codes is generalized from self-similar lattice codes to a broader class of lattices. If coding lattice $Lambda_{mathrm c}$ and shaping lattice $Lambda_{mathrm s}$ satisfy $Lambda _{mathrm s}subseteq Lambda _{mathrm c}$ , then $Lambda _{mathrm c}/ Lambda _{mathrm s}$ is a quotient group that can be used to form a (nested) lattice code $mathcal C$ . Conway and Sloane’s method of encoding and indexing does not apply when the lattices are not self-similar. Results are provided for two classes of lattices. 1) If $Lambda _{mathrm c}$ and $Lambda _{mathrm s}$ both have generator matrices in a triangular form that satisfies $Lambda _{mathrm s}subseteq Lambda _{mathrm c}$ , then encoding is always possible. 2) When $Lambda _{mathrm c}$ and $Lambda _{mathrm s}$ are described by full generator matrices, if a solution to a linear diophantine equation exists, then encoding is possible. In addition, special cases where $mathcal C$ is a cyclic code are considered. A condition for the existence of a group isomorphism between the information and $mathcal C$ is given. The results are applicable to a variety of coding lattices, including Construction A, Construction D, and low-density lattice codes. A variety of shaping lattices may be used as well, including convolutional code lattices and the direct sum of important lattices such as $D_{4}$ , $E_{8}$ , etc. Thus, a lattice code $mathcal C$ can be designed by selecting $Lambda _{mathrm c}$ and $Lambda _{mathrm s}$ separately, avoiding the competing design requirements of self-similar lattice codes." @default.
- W2963027968 created "2019-07-30" @default.
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- W2963027968 date "2018-09-01" @default.
- W2963027968 modified "2023-10-14" @default.
- W2963027968 title "Encoding and Indexing of Lattice Codes" @default.
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- W2963027968 doi "https://doi.org/10.1109/tit.2018.2839181" @default.
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