Towards an Algebraic Network Information Theory: Distributed Lossy Computation of Linear Functions

Consider the important special case of the K-user distributed source coding problem where the decoder only wishes to recover one or more linear combinations of the sources. The work of Körner and Marton demonstrated that, in some cases, the optimal rate region is attained by random linear codes, and strictly improves upon the best-known achievable rate region established via random i.i.d. codes. Recent efforts have sought to develop a framework for characterizing the achievable rate region for nested linear codes via joint typicality encoding and decoding. Here, we make further progress along this direction by proposing an achievable rate region for simultaneous joint typicality decoding of nested linear codes. Our approach generalizes the results of Körner and Marton to computing an arbitrary number of linear combinations and to the lossy computation setting.

[1]  Arun Padakandla,et al.  An Achievable Rate Region for the Three-User Interference Channel Based on Coset Codes , 2014, IEEE Transactions on Information Theory.

[2]  Abbas El Gamal,et al.  Network Information Theory , 2021, 2021 IEEE 3rd International Conference on Advanced Trends in Information Theory (ATIT).

[3]  Sung Hoon Lim,et al.  Optimal Achievable Rates for Computation With Random Homologous Codes , 2018, 2018 IEEE International Symposium on Information Theory (ISIT).

[4]  R. A. McDonald,et al.  Noiseless Coding of Correlated Information Sources , 1973 .

[5]  S.S. Pradhan,et al.  Distributed source coding using Abelian group codes: Extracting performance from structure , 2008, 2008 46th Annual Allerton Conference on Communication, Control, and Computing.

[6]  Aaron D. Wyner,et al.  Recent results in the Shannon theory , 1974, IEEE Trans. Inf. Theory.

[7]  KrithivasanD.,et al.  Distributed Source Coding Using Abelian Group Codes , 2011 .

[8]  Imre Csiszár Linear codes for sources and source networks: Error exponents, universal coding , 1982, IEEE Trans. Inf. Theory.

[9]  Arun Padakandla,et al.  Achievable rate region for three user discrete broadcast channel based on coset codes , 2012, 2013 IEEE International Symposium on Information Theory.

[10]  János Körner,et al.  How to encode the modulo-two sum of binary sources (Corresp.) , 1979, IEEE Trans. Inf. Theory.

[11]  Sui Tung,et al.  Multiterminal source coding (Ph.D. Thesis abstr.) , 1978, IEEE Trans. Inf. Theory.

[12]  Michael Gastpar,et al.  Towards an algebraic network information theory: Simultaneous joint typicality decoding , 2017, 2017 IEEE International Symposium on Information Theory (ISIT).

[13]  P. Vijay Kumar,et al.  Linear Coding Schemes for the Distributed Computation of Subspaces , 2013, IEEE Journal on Selected Areas in Communications.

[14]  Te Sun Han,et al.  A dichotomy of functions F(X, Y) of correlated sources (X, Y) , 1987, IEEE Trans. Inf. Theory.

[15]  Michael Gastpar,et al.  A Joint Typicality Approach to Compute–Forward , 2018, IEEE Transactions on Information Theory.

[16]  Young-Han Kim,et al.  Homologous codes for multiple access channels , 2017, 2017 IEEE International Symposium on Information Theory (ISIT).