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

Multiplicative matrix channel (MMC) over R

• Input:• Output:• transfer matrix:• Channel:

Field RingMotivated 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:

R A finite chain ring

π Any generator for the maximal ideal of R

s The nilpotency index of π (πs=0)

q The order of the residue field R/<π>

Γ Any set of coset representatives for R/<π>

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)

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.

Motivating example

Motivating example

Motivating example

In general, you (only) can compute

Roadmap

Channel Model

Channel Capacity Coding Scheme for One Shot, Coherent

Remark1) Code feature, extension to non-coherent

2) reliable communication for “shape deficiency” of the transfer matrix

Channel Modeln An integer (#transmitted packets)

m An integer (#received packets)

λ An s-shape (shape of packet space)

pA A probability distribution over Rmxn

Channel Model• Matrix Code• Codebook

(Multi-shot code/one-shot code)• Decoding function• Rate of the code• Channel Capacity

Channel Capacity

Proof Sketch of Theorem 3

Coding SchemeThe 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

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 codeSimilar 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/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.