The Story of (T,M,S)-Nets
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Transcript of The Story of (T,M,S)-Nets
The Story of (T,M,S)-Nets
Bill MartinMathematical Sciences and Computer ScienceWorcester Polytechnic Institute
Caveats, etc.
Many photos borrowed from the web (sources available on request)
This talk focuses only on the combinatorics; there is a lot more activity that I won’t talk about
WPI is looking for graduate students and visiting faculty
Mathematics Being Done in Many Places . . .
. . . By Many Kinds of People
. . . By Many Kinds of People
. . . By Many Kinds of People
. . . By Many Kinds of People
. . . By Many Kinds of People
. . . By Many Kinds of People
. . . By Many Kinds of People
. . . By Many Kinds of People
. . . By Many Kinds of People
. . . By Many Kinds of People
. . . By Many Kinds of People
. . . By Many Kinds of People
. . . By Many Kinds of People
. . . By Many Kinds of People
Pre-History
Quadrature rulesNumerical simulationGlobal optimization
Quasi-Random is not RandomRandomPseudo-random (should fool an
observer)Quasi-Random: entirely
deterministic, but has some statistical properties that a random set “should” have
Some Ways to Sample the CubeRandom (Monte Carlo)Lattice rulesLatin hypercube sampling (T,M,S)-nets
Evenly Sampling the Unit Cube A set N of N points inside [0,1)s
An interval E = [0,a1)x[0,a2)x . . . x[0,as)
“should” contain Vol(E) |N | of these points
The star discrepancy of a set N of N points in [0,1)s is the supremum of
| |N E| / N - Vol(E) |
taken over all such intervals E. Call it D*(N )
U
Koksma-Hlawka Inequality
J. Koksma E. Hlawka
Elementary Intervals
For any given shape (d1,d2,. . .,ds), the unit cube is partitioned into bm elementary intervals of this shape, each being a translate of every other.
Vienna, Austria 1980s
(T,M,S)-Nets
Harald Niederreiter
Working on low discrepancy sequences, quasi-randomness, pseudo-random generators, applications to numerical analysis, coding theory, cryptography
Expertise in finite fields and number theory
(T,M,S)-Nets
Niederreiter (1987), generalizing an idea of Sobol’ (1967)
Example
Sampling Evenly
Sampling Evenly
Sampling Evenly
Sampling Evenly
Sampling Evenly
Sampling Evenly
Sampling Evenly
Sampling Evenly
Sampling Evenly
Sampling Evenly
Sampling Evenly
Sampling Evenly
Using Latin Squares
Two MOLS(3) yield an orthogonal array of strength two
Latin Squares to (0,2,2)-net
Replace alphabet by {0,1,…,b-1} (here, base b=3)
Latin Squares to (0,2,2)-net
Insert decimal points to obtain a (0,2,2)-net in base 3
The Resulting (T,M,S)-Net
(0,2,2)-net in base 3
Su Doku!
Now fill in with cosets of the linear code
Vienna, Austria 1980sMadison, Wisconsin 1995
Generalized Orthogonal Arrays
Mark Lawrence, Chief Risk Officer, Australia and New Zealand Banking Group
Generalized Orthogonal Arrays In an orthogonal array of strength t, all entries are chosen from some fixed alphabet {0,1,. . .,b-1}
In any t columns, every possible t-tuple over the alphabet (there are qt of these) appears equally often So the total number of rows is l.bt where l is the replication number
If this hold for a set of columns, then it also holds for all subsets of that set
Now specify a partial order on the columns and require this only for lower ideals in this poset of size t or less
Vienna, Austria 1980sSalzburg, Austria 1995
Ordered Orthogonal Arrays
Wolfgang Ch. Schmid and Gary Mullen
Introduced OOA concept Proved equivalence to (T,M,S)-nets constructions bounds
OOA
Sample OOA from Simplex Code
0 0
0 0
0 0
0
0 0
1 0
1 1
1
1 0
0 1
0 1
1
1 1
0 0
1 0
1
1 1
1 0
0 1
0
0 1
1 1
0 0
1
1 0
1 1
1 0
0
0 1
0 1
1 1
0
Sample OOA1( 3, 3, 3, 2)
0 0 0
0 0 0
0 0 0
0 0 1
1 0 1
1 1 1
1 0 1
0 1 1
0 1 1
1 1 1
0 0 1
1 0 1
1 1 0
1 0 0
0 1 0
0 1 1
1 1 1
0 0 1
1 0 0
1 1 0
1 0 0
0 1 0
0 1 0
1 1 0
Schmid-Lawrence TheoremThere exists a (T,M,S)-net in base b
If and only if
there exists an OOAl( t, s, l, v)where
s=S t=l=M-T v=b l= bT
Proof Idea
Vienna, Austria 1980sSingapore 1995
Nets from Algebraic Curves
Harald Niederreiter and Chaoping Xing ( here pictured with Sang Lin)
Global function fields with many rational places
A Simpler Construction For simplicity, assume q is a prime
Let S = { p1, p2, . . . , ps} be a subset of Fq (or PG(1,q) )
Fix k >= 0 and create one point for each polynomial f(x) in Fq[x] of degree k or less
In the ith coordinate position, take f(pi)/q + f(1)(pi)/q2 + . . . + f(k)(pi)/qk+1
where f(j) denotes the jth derivative of f
A Simpler Construction To illustrate, let’s take
q = 5 k = 2 S = { 1, 2, 3} inside F5
For example, the polynomial f(x) = 3 x2 + 4 xhas f(1)(x) = x + 4 and f(2)(x) = 1
This contributes the point in [0,1)3
( .208, .048, .888 )
Example
First 5 points (constant polys)
Example
First 10 pts (constant &linear)
Example
First 15 points (constant & linear)
Example
First 20 points (constant & linear)
Example
First 25 points (all const & lin)
Example
First 50 points
Example
First 75 points
Example
First 100 points
Example – a (0,3,3)-net in base 5
All 125 points
Example – a (0,3,3)-net in base 5
All 125 points – another viewpoint
Vienna, Austria 1980s
Heidelberg, Germany 1995
Vienna, Austria 1980sHoughton, Michigan 1995
From Codes to Nets
Yves Edel and Juergen Bierbrauer
Digital nets from BCH codes . . . and twisted BCH codes
Vienna, Austria 1980sMoscow, Russia 1995
Codes for the m-Metric
M. Yu. Rosenbloom and Michael Tsfasman
Codewords are matrices Errors affect entire tail of a row algebraic geometry codes Gilbert-Varshamov bound . . . and more
Vienna, Austria 1980sAuburn Alabama 1995
How I got involved Auburn workshop in 1995 Reception at Pebble Hill Juergen Bierbrauer teaches me about (t,m,s)-nets over snacks Questions: “Is there a linear programming bound for these things?”
“Is there a MacWilliams-type theorem for duality?”
Vienna, Austria 1980sLaramie, Wyoming 1996
Vienna, Austria 1980sOutside Laramie
Poset Codes
Michael Adams Completed dissertation at U. Wyoming under Bryan Shader Poset metrics for codes New constructions of nets Convincing argument that MacWilliams identities DON’T exist
Vienna, Austria 1980sWinnipeg, Manitoba 1997
Vienna, Austria 1980sWinnipeg, March 1997
Vienna, Austria 1980s
University of Manitoba
Vienna, Austria 1980s
University of Nebraska
Generalized Rao Bound
Ordered Hamming Scheme
Doug Stinson and WJM
Self-dual association scheme generalising the Hamming schemes Duality between codes and OOAs MacWilliams identities, LP bound
Ordered Hamming Scheme
Ordered Hamming Scheme
How to Learn of New Results
Vladimir Levenshtein BCC at Queen Mary & Westfield College (qmul)“Look at this paper by Rosenbloom and Tsfasman”
RT Codes
RT Codes
Dual Codes and MacWilliams
Dual Codes and MacWilliams
MacWilliams Identity (Stinson/WJM)
Duality: RT codes and OOAs
St. Petersburg, Russia 1999
Uniform Distributions
Steven Dougherty and Maxim Skriganov
MDS Codes and Duality
Skriganov and then Dougherty/Skriganov: independently re-discovered a lot of the above MDS codes for the m-metric MacWilliams identities bounds and constructions
Houghton, Michigan
Vienna, Austria 1980sWinnipeg, Manitoba 1997
The Dual Plotkin Bound
Terry Visentin and WJM
Vienna, Austria 1980sSalzburg, Austria 1995
The State of the Art
Wolfgang Ch. Schmid and Rudi SchurerMany contributionsBut also a comprehensive on-line table of parameters with links to literature
Thank You
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