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: Now presented by Chun Lam Chan

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

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/

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)

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.

Motivating example

In general, you (only) can compute

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

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

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

Proof Sketch of Theorem 3

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.

Coding Scheme - Codebook Codebook For example,

Coding Scheme – Decoding Algo.

Basis for the Coding Scheme For 0≤i<s

Basis for the Coding Scheme

Lemma 5 Discarding unknowns Projecting into F

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]

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

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

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)

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/ , Apr 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.