On Symmetric Multilevel Diversity Coding System With Linear Computations

In a conventional Symmetric Multilevel Diversity Coding System (SMDCS), sources of varying importance are encoded by multiple encoders and a level-
$k$
decoder, if given access to any
$k$
encoders, can decode the first
$k$
most important sources. In practical large-scale applications like Non-terrestrial Networks, sinks (e.g., land earth stations) may be only interested in some functions of the sources (e.g., orbiting satellites) rather than the sources themselves. This letter introduces linear network function computations into the SMDCS, where a level-
$k$
decoder requests a linear function of the first
$k$
sources. The coding rate region of the 3-encoder (i.e., 3-level) SMDCS with linear computations (SMDCS-LC) was obtained. It is shown that when the coefficient vectors are dependent, coding between sources may help in reducing the information under transmission by encoders. However, when they are independent, superposition (i.e., sources-separated) coding is optimal. Furthermore, the minimum sum rate of the generic
$L$
-level SMDCS-LC is obtained. This can find applications like federated learning aggregation in Non-Terrestrial Networks.