On Multiplicative Matrix Channels over Finite Chain Rings

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: Field Ring Motivated by PNC Message space W being a finite T-module, where T is a PID A special situation commonly found in practice:

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-adicdecomposition:

4. Modules and Matrices over Chain Ring • **An s-shape μ=(μ0,μ1,…μs-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 Z8, shape A = (1,2,2) • **Let λ be an s-shape, Rnxλ is matrices with row constraints.

6. Motivating example

7. Motivating example

8. Motivating example In general, you (only) can compute

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

10. Channel Model

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

12. 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. Basis for the Coding Scheme Lemma 5 Discarding unknowns Projecting into F

20. Code Feature • Polynomial computational complexity (in m,n,l) • The rate of the code is given byThe 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.