  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.
