This document discusses lattice codes, which are error-correcting codes used in digital communication systems. Lattice codes use geometric structures called lattices that allow dense packing of signal points. This dense packing provides coding gain over other signaling methods. Specific lattices discussed include the Barnes-Wall lattices, Leech lattice, and trellis codes, which can be viewed as lattice codes with better complexity/performance. Key parameters for lattices include minimum distance, kissing number, and volume. Lattice codes map binary data to signal points in the lattice constellation and allow decoding by finding the closest lattice point.

1. LATTICE CODES
Samuel Cherukutty
208114018
M.Tech

2. Outline
Lattice Codes – An Intro
Lattices
Examples of Lattices
Geometrical Parameters of Lattice
Lattice Constellation
Shaping Gain
Coding Gain
Examples of Lattices used in Communication
Systems
Coding and Decoding with Lattices

3. Lattice Codes – An Intro
Why do we need more dimensions??
More freedom
More packing space
What is lattice?
a regular repeated three-dimensional arrangement of
atoms, ions, or molecules in a metal or other
crystalline solid.
What are the important criterions in defining
lattice??
Shape
Packing Density

4. Lattices(Math Perspective)
An n-dimensional (n-D) lattice Λ is a discrete
subset of n-space Rn that has the group
property.
Λ may be assumed to span Rn.The points of
the lattice then form a uniform infinite packing
of Rn.

5. Lattices (Examples)
Example 1. The set of integers Z is a one-dimensional lattice, since Z is
a discrete subgroup of R. Any 1-dimensional lattice is of the form Λ = αZ
for some scalar α > 0.
Example 2. The integer lattice Zn (the set of integer n-tuples) is an n-
dimensional lattice for any n ≥ 1.
Example 3. The hexagonal lattice A2 = {a(1, 0) + b( 1/2 ,√3/2)| (a,b) ∈ Z2

6. Geometric Parameters Of
Lattice
Minimum Squared Distance d2 between
lattice points
Vonori Region V of a lattice point is the set of
all points near to it in Rn
Kissing number Kmin(Λ) (the number of
nearest neighbors to any lattice point)
Volume V(Λ) of n-space per lattice point
Hermite Parameter(Normalized density
parameter)

7. Lattice Constellation
A lattice constellation
C(Λ, R) = (Λ + t) ∩ R
is the finite set of points in a lattice translate Λ + t that lie within
a compact bounding region R of n-space.

8. Shaping Gain
Given as
The n-dimensional shaping region R that
minimizes G(R) is obviously an n-sphere.
Ultimate Shaping Gain:- Shaping gain
reaches an ultimate value as n→ infinty =πe/6

9. Coding Gain
Can be increased by varying the constellations
Given by
Increases as n→ inf

10. Eg:Barnes-Wall lattices
Discovered in 1959
Infinite family of n-dim lattices analogous to
Reed-Muller binary block codes
Works good for n<16
Very good 'performance vs decoding
complexity'

11. Eg:Leech Lattice
This is one of the most famous lattices
Densest known packing in 24-dimensions.
All densest known lattice packings in fewer
than 24 dimensions occur as sections of Λ24.
It was discovered by Leech in 1965.
There are many constructions for it, but the
simplest one is by “induction”:
beginning with the densest 1-D lattice Z,
at each step extend to densest lattice in the next
dimension.
This construction, where from step i to i+1 one

12. Coding and Decoding
Coding
Select the appropriate Lattice considering type of
system,No: of Channels,etc
Derive the generating matrix from the vectors
spanning the Lattice
Use the matrix to code the data
Decoding
Inverse mapping of data
Finding the lattice closest to the inverse mapped
data

13. The Cyclic Brother:Trellis Code
Lattice Code Analogous to Trellis Code as
Linear Block Code to Cyclic Code
Provides better complexity/performance
tradeoff in comparison to Lattice codes
The key ideas in the invention of trellis codes
were:
use of minimum squared Euclidean distance as
the design criterion
coding on subsets of signal sets using
convolutional coding principles (e.g., trellises and
the Viterbi algorithm).

14. To Conclude
Lattice code consists of finite set of vectors
derived from Lattice
Binary info is mapped to the vectors in one-
one fashion
Lattice vector specifies amplitude of pulses in
M-PAM system
Encoding: Maping binary info to lattice vectos
Decoding: Finding the closest lattice on
performing inverse mapping
Kissing Number → no: of neighbours at free-distance
of code

15. References
Digital Communication by Simon Haykin
www.ocw.mit.edu/courses/electrical-
engineering-and-computer-science/6-451-
principles-of-digital-communication-ii-spring-
2005/
-MIT openCourse digital communication systems
by Prof. David Forney (lecture 22,23)
Digital Communication by LEE