Abstract
This paper presents a coding theorem for linear coding over finite rings, in the setting of the Slepian–Wolf source coding problem. This theorem covers corresponding achievability theorems of Elias (IRE Conv. Rec. 1955, 3, 37–46) and Csiszár (IEEE Trans. Inf. Theory 1982, 28, 585–592) for linear coding over finite fields as special cases. In addition, it is shown that, for any set of finite correlated discrete memoryless 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. Hence, the optimality problem regarding linear coding over finite non-field rings for data compression is closed with positive confirmation with respect to existence. For application, we address the problem of source coding for computing, where the decoder is interested in recovering a discrete function of the data generated and independently encoded by several correlated i.i.d. random sources. We propose linear coding over finite rings as an alternative solution to this problem. Results in Körner–Marton (IEEE Trans. Inf. Theory 1979, 25, 219–221) and Ahlswede–Han (IEEE Trans. Inf. Theory 1983, 29, 396–411, Theorem 10) are generalized to cases for encoding (pseudo) nomographic functions (over rings). Since a discrete function with a finite domain always admits a nomographic presentation, we conclude that both generalizations universally apply for encoding all discrete functions of finite domains. Based on these, we demonstrate that linear coding over finite rings strictly outperforms its field counterpart in terms of achieving better coding rates and reducing the required alphabet sizes of the encoders for encoding infinitely many discrete functions.
Highlights
The problem of source coding for computing considers the scenario where a decoder is interested in recovering a function of the message(s), other than the original message(s), that is i.i.d. generated and independently encoded by the source(s)
We have only shown that there exist linear encoders over finite non-field rings that are good as their field counterparts
Lemma 4 implies that any discrete function defined on a finite domain is equivalent to a restriction of some polynomial function over some finite ring
Summary
The problem of source coding for computing considers the scenario where a decoder is interested in recovering a function of the message(s), other than the original message(s), that is (are) i.i.d. generated and independently encoded by the source(s). It is proved that there always exists a sequence of linear encoders over some finite non-field rings that achieves the SW region for any scenario of SW. Wedderburn’s little theorem guarantees commutativity for a finite ring if all of its non-zero elements are invertible. We emphasize that even if group, field and ring are closely related algebraic structures, the definition of the group encoder in [11] and the linear encoder in [3] and in the present work are in general fundamentally different ( there is an overlap in special cases). In this quite specific special case, the encoder becomes linear over a product ring of modulo integers, which is a sub-class of the completely general ring structures considered in this paper.
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