Lattice index codes from algebraic number fields

Abstract

Broadcasting K independent messages to multiple users where each user demands all the messages and has a subset of the messages as side information is studied. Recently, Natarajan, Hong, and Viterbo proposed a novel broadcasting strategy called lattice index coding which uses lattices constructed over some principal ideal domains (PIDs) for transmission and showed that this scheme provides uniform side information gains. In this paper, we generalize this strategy to general rings of algebraic integers of number fields which may not be PIDs. Upper and lower bounds on the side information gains for the proposed scheme constructed over some interesting classes of number fields are provided and are shown to coincide asymptotically in message rates. This generalization substantially enlarges the design space and partially includes the scheme by Natarajan, Hong, and Viterbo as a special case. Perhaps more importantly, in addition to side information gains, the proposed lattice index codes benefit from diversity gains inherent in constellations carved from number fields when used over Rayleigh fading channel. Some interesting examples are also provided for which the proposed scheme allows all the messages to be from the same field.I. INTRODUCTION Broadcast with receiver message side information has recently attracted a lot of attention in the network layer with many interesting and important advances. For example, in the index coding problem [1] [2] [3], a sender broadcasts a set of independent messages to several receivers where each receiver demands a subset of messages and has another subset of messages as side information. Another excellent example is the caching problem [4] where a sender broadcasts a set of independent messages with a fraction of messages (or functions of messages) being prefetched into receivers beforehand during off-peak hours. For such problems, it has been shown that a carefully designed broadcasting strategy which allows the receivers to better exploit side information can substantially improve the system throughput.As the physical layer counterpart, Gaussian broadcast channels with receiver side information have recently been popular as well. In [5], the two-user Gaussian broadcast channel with receiver message side information was studied and the capacity region was fully characterized for all (5 in total) possible side information configurations. Yoo et al. in [6] considered the three-user scenario where they showed that a separation-based scheme which separately employs index coding and physical layer coding can achieve the capacity region to within a constant gap for all side information configurations regardless of channel parameters. For the three-user case, the capacity region was characterized for some particular side information configurations [7] [8]. For the case having more than three users, our knowledge is fairly limited. This problem has also been independently studied in the context of the broadcast phase of the two-way (or multi-way) relay channel [9].Recently, in ...

On the design of multi-dimensional irregular repeat-accumulate lattice codes

“…In this way, the D 4 lattice is partitioned into 25 cosets. Even though H is a PID [36], we only have the group homomorphism as the multiplication for H is non-commutative. Now we compare the mutual information of the D 4 lattice with that of a two-dimensional lattice to see the performance gain introduced by the multi-dimensional lattices.…”

Layered Space-Time Index Coding

“…In [17], Huang considers the same multicasting problem with message side information, where each link experiences a Rayleigh fading channel on top of the AWGN noise. It is well-known that in contrast to the AWGN channel, maximizing minimum distance alone is far from enough for the Rayleigh fading channel and the minimum product distance dominates the performance [19]- [21].…”

“…It is well-known that in contrast to the AWGN channel, maximizing minimum distance alone is far from enough for the Rayleigh fading channel and the minimum product distance dominates the performance [19]- [21]. The lattice index coding scheme proposed in [17] generalizes the idea of [16] from some famous principal ideal domains to any ring of algebraic integers. It is shown that codes thus constructed over rings of algebraic integers of totally real number fields provide gains in minimum product distance that is exponential with the rate of the side information for any index set.…”

“…For such a MIMO setting, the minimum determinant of the code serves as one of the most important parameters to be maximized [21], [22] and algebraic space-time block codes (STBC) constructed from cyclic division algebras [23]- [26] are a class of codes that possess many desired properties. Since CRT does not hold for non-commutative rings such as cyclic division algebras, the trick used in [16] and [17] does not work here in general. In [18], the problem is circumvented by using the bijective mapping between the Golden algebra and a commutative ring found in [27] together with some special ideals whose group structure is preserved by the mapping.…”

Golden-coded index coding

“…In [6], the same problem was studied for the Rayleigh fading channel in which the minimum product distance is much more important than the minimum Euclidean distance. The new lattice index code construction for the Rayleigh fading channel in [6] provides exponentially increased squared minimum product distance as the rate of side information increases and provides uniform side information gain for any side information configuration.…”