ISSN 0032-9460, Problems of Information Transmission, 2017, Vol. 53, No. 1, pp. 1–29.
Pleiades Publishing, Inc., 2017.
Original Russian Text
S. Vatedka, N. Kashyap, 2017, published in Problemy Peredachi Informatsii, 2017, Vol. 53, No. 1, pp. 3–33.
Some “Goodness” Properties of LDA Lattices
and N. Kashyap
Department of Electrical Communication Engineering,
Indian Institute of Science, Bengaluru, India
Received December 14, 2015; in ﬁnal form, July 26, 2016
Abstract—We study some structural properties of Construction-A lattices obtained from low
density parity check codes over prime ﬁelds. Such lattices are called low density Construction-A
(LDA) lattices, and permit low-complexity belief propagation decoding for transmission over
Gaussian channels. It has been shown that LDA lattices achieve the capacity of the power
constrained additive white Gaussian noise (AWGN) channel with closest lattice-point decod-
ing, and simulations suggested that they perform well under belief propagation decoding. We
continue this line of work and prove that these lattices are good for packing and mean squared
error quantization and that their duals are good for packing. With this, we can conclude that
codes constructed using nested LDA lattices can achieve the capacity of the power constrained
AWGN channel, the capacity of the dirty paper channel, the rates guaranteed by the compute-
and-forward protocol, and the best known rates for bidirectional relaying with perfect secrecy.
Nested lattice coding for communication over Gaussian networks has received considerable at-
tention in recent times. It has been shown  that nested lattice codes with closest lattice-point
decoding can achieve the capacity of the power constrained additive white Gaussian noise (AWGN)
channel. They are also known to achieve the capacity of the dirty-paper channel . Inspired by
these results, they have been applied to design protocols for reliable communication over wireless
Gaussian networks. They have been used with much success for the interference channel [3, 4], the
Gaussian bidirectional relay channel [5,6], and generalized to the problem of physical layer network
coding [6, 7] for multiuser Gaussian channels. Nested lattice coding has also been used for secu-
rity in wiretap channels [8,9] and bidirectional relay networks [10, 11]. For a more comprehensive
treatment of lattices and their applications in communication problems, see .
Constructing lattices that have good structural properties is a problem that has been studied for
a long time. Poltyrev  studied lattices in the context of coding for reliable transmission over the
AWGN channel without power constraints and showed that there exist lattices which are “good”
for AWGN channel coding, i.e., achieve a vanishingly small probability of error for all suﬃciently
small values of the noise variance. In addition to coding for the AWGN channel, lattices were also
studied in prior literature in the context of several other problems such as sphere packing, sphere
covering, and mean squared error (MSE) quantization. In the sphere packing problem, we want to
ﬁnd an arrangement of nonintersecting spheres of a given radius that maximizes the average number
This work was presented in part at the 2015 IEEE Information Theory Workshop at Jerusalem, Israel.
Supported in part by the Tata Consultancy Services Research Scholarship Program.
Supported in part by a Swarnajayanti fellowship awarded by the Department of Science and Technology,