Latticecoded

Lattice-coded refers to communication schemes that employ lattice codes—structured point sets in Euclidean space—as the basis for encoding and decoding information. A lattice is a discrete grid formed by all integer combinations of a set of basis vectors in n-dimensional space. A lattice code is obtained by selecting a finite subset of lattice points, often within a shaping region, or by using a nested pair of lattices to separate shaping from coding. Standard construction methods, such as Construction A, D, and D', build lattice codes from linear codes over finite fields, enabling practical implementations.

Key ideas in lattice coding include shaping and coding gains. Shaping restricts transmitted power by confining code points to a region with favorable energy properties, commonly using the Voronoi region of a coarse lattice. The minimum distance of the lattice provides a natural measure of error protection, yielding a coding gain over unstructured codes. Decoding often relies on nearest-neighbor (maximum-likelihood) approaches; due to high dimensionality, practical schemes use sphere decoding or iterative algorithms, with complexity increasing with dimension.

In Gaussian channels, lattice codes can be provably capacity-achieving when combined with appropriate shaping and dithering, and nested lattice codes enable efficient implementation for multi-user settings. Lattice coding is central to topics such as lattice-coded modulation, compute-and-forward, and physical-layer network coding, where lattice structure facilitates interference management and multi-user encoding.

Lattice-coded schemes are distinct from, though related to, lattice-based cryptography, which uses lattices for security primitives rather than communication coding. Lattice coding continues to be explored for high-rate wireless systems, data compression, and networked communications.