Golden-coded index coding

Abstract

Abstract-We study the problem of constructing good spacetime codes for broadcasting K independent messages over a MIMO network to L users, where each user demands all the messages and already has a subset of messages as side information. As a first attempt, we consider the 2 × 2 case and propose golden-coded index coding by partitioning the golden codes into K subcodes, one for each message. The proposed scheme is shown to have the property that for any side information configuration, the minimum determinant of the code increases exponentially with the amount of information contained in the side information.

Layered Space-Time Index Coding

“…We note that the technique used in [18] requires the code to be constructed over prime ideals whose group structure are preserved by the bijective mapping of [27] and thereby limits the possible message sizes. In contrast to the Golden-coded index coding in [18] working only for Golden code with the message sizes confined to some particular prime powers, the proposed LSTIC is quite general in the following senses: 1) it works for a large class of STBC constructed from cyclic division algebras; and 2) it has less restriction on the message size. We would like to emphasize here that when specialized to the Golden code, the proposed LSTIC is not a special case of the Golden-coded index coding in [18] and vice versa, due to the different message sizes.…”

“…The second category including [14]- [18] considers designing codes/constellations that possess some desired properties in the finite dimension regime. The main objective is to design codes such that the probability of error will decrease by an amount that is proportional to the amount of information contained in the side information.…”

“…The objective then becomes designing coding schemes that are fair to every possible index set. As a starting point, only the multicasting case is considered in [15]- [18] where all the receivers demand all the messages.…”

“…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. The multicasting problem with message side information is then considered in [18] under the 2×2 MIMO setting where the transmitter and the receivers are equipped with two antennas. 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. As a result, we successfully construct Goldencoded index coding from Golden code, a subclass of perfect codes for the case with two transmitter and receiver antennas, and show that the minimum determinant increases exponentially with the rate of the side information for any index set.…”