Low-Density Lattice Codes

Abstract

Abstract-Low-density lattice codes (LDLC) are novel lattice codes that can be decoded efficiently and approach the capacity of the additive white Gaussian noise (AWGN) channel. In LDLC a codeword x is generated directly at the n-dimensional Euclidean space as a linear transformation of a corresponding integer message vector b, i.e., x = G G Gb, where H H H = G G G 01 is restricted to be sparse. The fact that H H H is sparse is utilized to develop a linear-time iterative decoding scheme which attains, as demonstrated by simulations, good error performance within 0.5 dB from capacity at block length of n =100,000 symbols. The paper also discusses convergence results and implementation considerations.

Construction of turbo lattices

“…For lattices, instead of coding rate and capacity, we use normalized logarithmic density and generalized capacity. In this subsection, we recall these parameters from [9,14,18].…”

“…The above representation (18) can be extended to construct a generator matrix for parallel concatenated codes with b branches. Each branch has its own interleaver Π j with matrix representation P j and a recursive encoder E j for 1 ≤ j ≤ b.…”

“…, k 1 )-nested property that is used to construct a turbo code TC. Then the generator matrix G T C is as of (18). Define a set of nested matrices…”

“…There has been interest in constructing lattices with high encoding efficiency and low decoding complexity [1,15,18]. It is known [9,20] that we can send points of a lattice over an AWGN channel with noise variance σ 2 and then recover the received points by means of an appropriate lattice decoding algorithm.…”

“…There are a wide range of applicable lattices in communications including the well-known root lattices [6], the recently introduced low density parity check lattices [15] (LDPC lattices) and the low density lattice codes [18] (LDLC lattices). The former lattices have been extensively treated in the 1980's and 1990's [6].…”