Lattice Index Coding

Abstract

Abstract-The index coding problem involves a sender with K messages to be transmitted across a broadcast channel, and a set of receivers each of which demands a subset of the K messages while having prior knowledge of a different subset as side information. We consider the specific case of noisy index coding where the broadcast channel is Gaussian and every receiver demands all the messages from the source. Instances of this communication problem arise in wireless relay networks, sensor networks, and retransmissions in broadcast channels. We construct lattice index codes for this channel by encoding the K messages individually using K modulo lattice constellations and transmitting their sum modulo a coarse lattice. We introduce a design metric called side information gain that measures the advantage of a code in utilizing the side information at the receivers, and hence, its goodness as an index code. Based on the Chinese remainder theorem, we then construct lattice index codes with large side information gains using lattices over the following principal ideal domains: rational integers, Gaussian integers, Eisenstein integers, and the Hurwitz quaternions. Among all lattice index codes constructed using any densest lattice of a given dimension, our codes achieve the maximum side information gain. Finally, using an example, we illustrate how the proposed lattice index codes can benefit Gaussian broadcast channels with more general message demands.

Golden-coded index coding

“…For this case, the problem has been previously studied for the AWGN channel [5], where a new class of codes named lattice index codes based on lattice codes is proposed to mimic the behavior of capacity-achieving codes. The lattice index codes in [5] are shown to have the minimum squared Euclidean distance increasing exponentially as the rate of side information for any side information configuration. Moreover, when normalized by the rate of side information, the SNR difference between the codes with and without side information for achieving the same error probability is 6 dB/bit.…”

“…While there is a rich literature in the study of construction of space-time codes for the point-to-point MIMO channel (see [7] and the reference therein), as a first attempt, we consider construction of lattice space-time index codes solely based on golden codes [8] for the 2 × 2 case. The main difficulty is that most of the code constructions proposed in [5] and [6] rely on partitions induced by the Chinese remainder theorem (CRT) for some commutative rings; however, golden codes (and most of the lattice space-time codes) are constructed over a cyclic division algebra, which is noncommutative and hence prevents the direct application of CRT. We overcome this challenge and propose the golden-coded index coding by making connection between the underlying cyclic division algebra and a ring of algebraic integers and then partitioning this ring instead.…”

Lattice index codes from algebraic number fields

“…Recently, in [10], Natarajan et al considered a special class of Gaussian broadcast channels with receiver message side information as shown in Fig. 1.…”

“…In this model, each receiver demands all the messages and can have an arbitrary subset of messages as side information. Unlike other work focusing on the capacity region, [10] focused on designing practical codes/modulations and proposed the so-called lattice index codes that provide large side information gains for any side information configuration. In [11], they further proposed the index coded modulation which adopts powerful linear codes as outer codes in conjunction with index modulation as inner code to enjoy coding gains on top of side information gains.…”

“…In this paper, we study the same special class of Gaussian broadcast channels with receiver message side information as that in [10] and also focus on practical code/modulation design. The contributions of this paper is summarized in the following:…”