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.
