1. On Multiplicative Matrix Channels over Finite Chain Rings Roberto W. Nobrega, Chen Feng, Danilo Silva, Bartolomeu F. Uchoa-Filho Conference version: NetCod 2013 Journal version: preprint at arXiv:1311.4861 Now presented by Chun Lam Chan

2. Multiplicative matrix channel (MMC) over R Input: Output: transfer matrix: Channel: FieldRing Motivated by PNC A special situation commonly found in practice: Message space W being a finite T-module, where T is a PID

3. Finite Chain Ring A chain ring is a ring in which the ideals are linearly ordered under subset inclusion. **R has precisely s+1 ideals, namely, The π-adic decomposition: RA finite chain ring πAny generator for the maximal ideal of R sThe nilpotency index of π (π s =0) qThe order of the residue field R/ ΓAny set of coset representatives for R/

4. Modules and Matrices over Chain Ring **An s-shape μ=(μ 0,μ 1,…μ s-1 ) is a non-decreasing sequence of s non-negative integers. We define If M is a finite R-module, then for some unique s-shape μ. We write (~dimension of vector space) The shape of a matrix A is defined as (~rank)

5. Modules and Matrices over Chain Ring Let Recall the Smith normal form of A is For example, consider matrix A over Z 8, shape A = (1,2,2) **Let λ be an s-shape, R nxλ is matrices with row constraints.

6. Motivating example

8. In general, you (only) can compute

9. Roadmap Channel Model Channel CapacityCoding Scheme for One Shot, Coherent Remark 1)Code feature, extension to non-coherent 2)reliable communication for “shape deficiency” of the transfer matrix

10. Channel Model nAn integer (#transmitted packets) mAn integer (#received packets) λAn s-shape (shape of packet space) pApA A probability distribution over R mxn

11. Channel Model Matrix Code Codebook (Multi-shot code/one-shot code) Decoding function Rate of the code Channel Capacity

13. Proof Sketch of Theorem 3

14. Coding Scheme The residue field The natural projection map The coset representative selector Composite code: Combine s codes over the residue field to obtain a code over the chain ring.

15. Coding Scheme - Codebook Codebook For example,

16. Coding Scheme – Decoding Algo.

17. Basis for the Coding Scheme For 0≤i<s

18. Basis for the Coding Scheme

19. Lemma 5 Discarding unknowns Projecting into F

20. Code Feature Polynomial computational complexity (in m,n,l) The rate of the code is given by The error probability is upper bounded as Universality: The complete knowledge of the probability distribution of A is not needed, only the knowledge of E[ρ]

21. Extension to the Non-Coherent Scenario Prepend headers The overhead can be made negligible if we are allowed to arbitrarily increase the packet length

22. One-shot Reliable Communication MRD code = maximum rank distance code Similar to MDS (maximum distance separable) code

23. Opening Questions Capacity of non-coherent MMC in finite chain rings Design of capacity-achieving coding schemes for non- coherent MMC with small λ (in both finite chain ring case and finite-field case)

24. Reference S. Yang, S.-W. Ho, J. Meng, E.-h. Yang, and R. W. Yeung, “Linear operator channels over finite fields,” Computing Research Repository (CoRR), vol. abs/1002.2293, Apr. 2010. C. Feng, R. W. Nobrega, F. R. Kschischang, and D. Silva, “Communication over finite-ring matrix channels,” in Proceedings of the 2013 IEEE International Symposium on Information Theory (ISIT’13), (Istanbul, Turkey), July 2013.