On Multiplicative Matrix Channels over Finite Chain Rings Roberto W. Nobrega, Chen Feng, Danilo...
-
Upload
rafe-turner -
Category
Documents
-
view
218 -
download
0
Transcript of On Multiplicative Matrix Channels over Finite Chain Rings Roberto W. Nobrega, Chen Feng, Danilo...
![Page 1: On Multiplicative Matrix Channels over Finite Chain Rings Roberto W. Nobrega, Chen Feng, Danilo Silva, Bartolomeu F. Uchoa-Filho Conference version: NetCod.](https://reader035.fdocuments.in/reader035/viewer/2022062314/56649e715503460f94b6fc94/html5/thumbnails/1.jpg)
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
![Page 2: On Multiplicative Matrix Channels over Finite Chain Rings Roberto W. Nobrega, Chen Feng, Danilo Silva, Bartolomeu F. Uchoa-Filho Conference version: NetCod.](https://reader035.fdocuments.in/reader035/viewer/2022062314/56649e715503460f94b6fc94/html5/thumbnails/2.jpg)
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
![Page 3: On Multiplicative Matrix Channels over Finite Chain Rings Roberto W. Nobrega, Chen Feng, Danilo Silva, Bartolomeu F. Uchoa-Filho Conference version: NetCod.](https://reader035.fdocuments.in/reader035/viewer/2022062314/56649e715503460f94b6fc94/html5/thumbnails/3.jpg)
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>
![Page 4: On Multiplicative Matrix Channels over Finite Chain Rings Roberto W. Nobrega, Chen Feng, Danilo Silva, Bartolomeu F. Uchoa-Filho Conference version: NetCod.](https://reader035.fdocuments.in/reader035/viewer/2022062314/56649e715503460f94b6fc94/html5/thumbnails/4.jpg)
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)
![Page 5: On Multiplicative Matrix Channels over Finite Chain Rings Roberto W. Nobrega, Chen Feng, Danilo Silva, Bartolomeu F. Uchoa-Filho Conference version: NetCod.](https://reader035.fdocuments.in/reader035/viewer/2022062314/56649e715503460f94b6fc94/html5/thumbnails/5.jpg)
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.
![Page 6: On Multiplicative Matrix Channels over Finite Chain Rings Roberto W. Nobrega, Chen Feng, Danilo Silva, Bartolomeu F. Uchoa-Filho Conference version: NetCod.](https://reader035.fdocuments.in/reader035/viewer/2022062314/56649e715503460f94b6fc94/html5/thumbnails/6.jpg)
Motivating example
![Page 7: On Multiplicative Matrix Channels over Finite Chain Rings Roberto W. Nobrega, Chen Feng, Danilo Silva, Bartolomeu F. Uchoa-Filho Conference version: NetCod.](https://reader035.fdocuments.in/reader035/viewer/2022062314/56649e715503460f94b6fc94/html5/thumbnails/7.jpg)
Motivating example
![Page 8: On Multiplicative Matrix Channels over Finite Chain Rings Roberto W. Nobrega, Chen Feng, Danilo Silva, Bartolomeu F. Uchoa-Filho Conference version: NetCod.](https://reader035.fdocuments.in/reader035/viewer/2022062314/56649e715503460f94b6fc94/html5/thumbnails/8.jpg)
Motivating example
In general, you (only) can compute
![Page 9: On Multiplicative Matrix Channels over Finite Chain Rings Roberto W. Nobrega, Chen Feng, Danilo Silva, Bartolomeu F. Uchoa-Filho Conference version: NetCod.](https://reader035.fdocuments.in/reader035/viewer/2022062314/56649e715503460f94b6fc94/html5/thumbnails/9.jpg)
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
![Page 10: On Multiplicative Matrix Channels over Finite Chain Rings Roberto W. Nobrega, Chen Feng, Danilo Silva, Bartolomeu F. Uchoa-Filho Conference version: NetCod.](https://reader035.fdocuments.in/reader035/viewer/2022062314/56649e715503460f94b6fc94/html5/thumbnails/10.jpg)
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
![Page 11: On Multiplicative Matrix Channels over Finite Chain Rings Roberto W. Nobrega, Chen Feng, Danilo Silva, Bartolomeu F. Uchoa-Filho Conference version: NetCod.](https://reader035.fdocuments.in/reader035/viewer/2022062314/56649e715503460f94b6fc94/html5/thumbnails/11.jpg)
Channel Model• Matrix Code• Codebook
(Multi-shot code/one-shot code)• Decoding function• Rate of the code• Channel Capacity
![Page 12: On Multiplicative Matrix Channels over Finite Chain Rings Roberto W. Nobrega, Chen Feng, Danilo Silva, Bartolomeu F. Uchoa-Filho Conference version: NetCod.](https://reader035.fdocuments.in/reader035/viewer/2022062314/56649e715503460f94b6fc94/html5/thumbnails/12.jpg)
Channel Capacity
![Page 13: On Multiplicative Matrix Channels over Finite Chain Rings Roberto W. Nobrega, Chen Feng, Danilo Silva, Bartolomeu F. Uchoa-Filho Conference version: NetCod.](https://reader035.fdocuments.in/reader035/viewer/2022062314/56649e715503460f94b6fc94/html5/thumbnails/13.jpg)
Proof Sketch of Theorem 3
![Page 14: On Multiplicative Matrix Channels over Finite Chain Rings Roberto W. Nobrega, Chen Feng, Danilo Silva, Bartolomeu F. Uchoa-Filho Conference version: NetCod.](https://reader035.fdocuments.in/reader035/viewer/2022062314/56649e715503460f94b6fc94/html5/thumbnails/14.jpg)
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.
![Page 15: On Multiplicative Matrix Channels over Finite Chain Rings Roberto W. Nobrega, Chen Feng, Danilo Silva, Bartolomeu F. Uchoa-Filho Conference version: NetCod.](https://reader035.fdocuments.in/reader035/viewer/2022062314/56649e715503460f94b6fc94/html5/thumbnails/15.jpg)
Coding Scheme - Codebook• Codebook
For example,
![Page 16: On Multiplicative Matrix Channels over Finite Chain Rings Roberto W. Nobrega, Chen Feng, Danilo Silva, Bartolomeu F. Uchoa-Filho Conference version: NetCod.](https://reader035.fdocuments.in/reader035/viewer/2022062314/56649e715503460f94b6fc94/html5/thumbnails/16.jpg)
Coding Scheme – Decoding Algo.
![Page 17: On Multiplicative Matrix Channels over Finite Chain Rings Roberto W. Nobrega, Chen Feng, Danilo Silva, Bartolomeu F. Uchoa-Filho Conference version: NetCod.](https://reader035.fdocuments.in/reader035/viewer/2022062314/56649e715503460f94b6fc94/html5/thumbnails/17.jpg)
Basis for the Coding Scheme
For 0≤i<s
![Page 18: On Multiplicative Matrix Channels over Finite Chain Rings Roberto W. Nobrega, Chen Feng, Danilo Silva, Bartolomeu F. Uchoa-Filho Conference version: NetCod.](https://reader035.fdocuments.in/reader035/viewer/2022062314/56649e715503460f94b6fc94/html5/thumbnails/18.jpg)
Basis for the Coding Scheme
![Page 19: On Multiplicative Matrix Channels over Finite Chain Rings Roberto W. Nobrega, Chen Feng, Danilo Silva, Bartolomeu F. Uchoa-Filho Conference version: NetCod.](https://reader035.fdocuments.in/reader035/viewer/2022062314/56649e715503460f94b6fc94/html5/thumbnails/19.jpg)
Basis for the Coding Scheme
Lemma 5
Discarding unknowns
Projecting into F
![Page 20: On Multiplicative Matrix Channels over Finite Chain Rings Roberto W. Nobrega, Chen Feng, Danilo Silva, Bartolomeu F. Uchoa-Filho Conference version: NetCod.](https://reader035.fdocuments.in/reader035/viewer/2022062314/56649e715503460f94b6fc94/html5/thumbnails/20.jpg)
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]
![Page 21: On Multiplicative Matrix Channels over Finite Chain Rings Roberto W. Nobrega, Chen Feng, Danilo Silva, Bartolomeu F. Uchoa-Filho Conference version: NetCod.](https://reader035.fdocuments.in/reader035/viewer/2022062314/56649e715503460f94b6fc94/html5/thumbnails/21.jpg)
Extension to the Non-Coherent Scenario
• Prepend headers• The overhead can be made negligible if we are allowed to
arbitrarily increase the packet length
![Page 22: On Multiplicative Matrix Channels over Finite Chain Rings Roberto W. Nobrega, Chen Feng, Danilo Silva, Bartolomeu F. Uchoa-Filho Conference version: NetCod.](https://reader035.fdocuments.in/reader035/viewer/2022062314/56649e715503460f94b6fc94/html5/thumbnails/22.jpg)
One-shot Reliable Communication
MRD code = maximum rank distance codeSimilar to MDS (maximum distance separable) code
![Page 23: On Multiplicative Matrix Channels over Finite Chain Rings Roberto W. Nobrega, Chen Feng, Danilo Silva, Bartolomeu F. Uchoa-Filho Conference version: NetCod.](https://reader035.fdocuments.in/reader035/viewer/2022062314/56649e715503460f94b6fc94/html5/thumbnails/23.jpg)
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)
![Page 24: On Multiplicative Matrix Channels over Finite Chain Rings Roberto W. Nobrega, Chen Feng, Danilo Silva, Bartolomeu F. Uchoa-Filho Conference version: NetCod.](https://reader035.fdocuments.in/reader035/viewer/2022062314/56649e715503460f94b6fc94/html5/thumbnails/24.jpg)
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.