On existence of optimal linear encoders over non-field rings for data compression with application to computing

This note proves that, for any finite set of correlated discrete i.i.d. sources, there always exists a sequence of linear encoders over some finite non-field rings which achieves the data compression limit, the Slepian-Wolf region. Based on this, we address a variation of the data compression problem which considers recovering some discrete function of the data. It is demonstrated that linear encoder over non-field ring strictly outperforms its field counterpart for encoding some function in terms of achieving strictly larger achievable region with strictly smaller alphabet size.