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One open problem in source coding is to characterize the limits ofrepresenting losslessly a non-identity discrete function of the data encodedindependently by the encoders of several correlated sources with memory. Thispaper investigates this problem under Markovian conditions, namely either thesources or the functions considered are Markovian. We propose using linearmappings over finite rings as encoders. If the function considered admitscertain polynomial structure, the linear encoders can make use of thisstructure to establish "implicit collaboration" and boost the performance. Infact, this approach universally applies to any scenario (arbitrary function)because any discrete function admits a polynomial presentation of requiredformat.  There are several useful discoveries in the paper. The first says that linearencoder over non-field ring can be equally optimal for compressing datagenerated by an irreducible Markov source. Secondly, regarding the previousfunction-encoding problem, there are infinitely many circumstances where linearencoder over non-field ring strictly outperforms its field counterpart. To bemore precise, it is seen that the set of coding rates achieved by linearencoder over certain non-field rings is strictly larger than the one achievedby the field version, regardless which finite field is considered. Therefore,in this sense, linear coding over finite field is not optimal. In addition, forcertain scenarios where the sources do not possess the ergodic property, ourring approach is still able to offer a solution.

Coding for Computing Irreducible Markovian Functions of Sources with Memory