Distributed Source Coding Using Abelian Group Codes: A New Achievable Rate-Distortion Region

Krithivasan, Dinesh; Pradhan, S. Sandeep

doi:10.1109/tit.2010.2103852

Cited by 66 publications

(62 citation statements)

References 42 publications

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“…However, 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 (although there is an overlap in special cases). To highlight in more detail the difference between linear encoding (this work and [3]) and encoding over a group, as in [11], which is a nonlinear operation in general, take the Abelian group G = Z 2 ⊕ Z 2 , the field F 4 of order 4 and the matrix ring M L,2 = a 0 b a a, b ∈ Z 2 as examples.…”

Section: Remarkmentioning

confidence: 83%

“…It should be noted that an interesting approach to coding over an Abelian group was presented in [9][10][11]. However, 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 (although there is an overlap in special cases).…”

Section: Remarkmentioning

confidence: 83%

“…By ( [11], Example 2), the Abelian group encoder encodes the sourceẐ = (X, Y) ∈ G based on a Slepian-Wolf like scheme. Namely, two binary linear encoders encode X n and Y n separately as two binary sources.…”

Section: Remarkmentioning

confidence: 99%

“…For instance, for encoding the modulo-two sum of binary symmetric sources, linear coding over F 4 or M L,2 achieves the optimal Körner-Marton region [4] (the M L,2 case will be established in later sections), while coding over G achieves the sub-optimal Slepian-Wolf region ( [11], p. 1509). To avoid any remaining confusion, we in Appendix D present additional details regarding the differences between linear coding, as in the present work and in [3], and coding over an Abelian group, as in [11].…”

Section: Remarkmentioning

confidence: 99%

“…Conditions under which R[g] is strictly larger than the SW region are presented in [6,8] from different perspectives, respectively. The cases regarding Abelian group codes are covered in [9][10][11]. The present work proposes replacing the linear encoders over finite fields from Elias [2] and Csiszár [3] with linear encoders over finite rings in the case of the problems accounted for above.…”

Section: Introductionmentioning

confidence: 99%

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On Linear Coding over Finite Rings and Applications to Computing

Huang

Skoglund

2017

Entropy

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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.

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