the story of (t,m,s)-nets
DESCRIPTION
Bill Martin Mathematical Sciences and Computer Science Worcester Polytechnic Institute. The Story of (T,M,S)-Nets. Caveats, etc. Many photos borrowed from the web (sources available on request) - PowerPoint PPT PresentationTRANSCRIPT
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The Story of (T,M,S)-Nets
Bill MartinMathematical Sciences and Computer ScienceWorcester Polytechnic Institute
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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
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Mathematics Being Done in Many Places . . .
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. . . By Many Kinds of People
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. . . By Many Kinds of People
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. . . By Many Kinds of People
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. . . By Many Kinds of People
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. . . By Many Kinds of People
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. . . By Many Kinds of People
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. . . By Many Kinds of People
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. . . By Many Kinds of People
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. . . By Many Kinds of People
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. . . By Many Kinds of People
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. . . By Many Kinds of People
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. . . By Many Kinds of People
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. . . By Many Kinds of People
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. . . By Many Kinds of People
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Pre-History
Quadrature rulesNumerical simulationGlobal optimization
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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
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Some Ways to Sample the CubeRandom (Monte Carlo)Lattice rulesLatin hypercube sampling (T,M,S)-nets
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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
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Koksma-Hlawka Inequality
J. Koksma E. Hlawka
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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.
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Vienna, Austria 1980s
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(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
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(T,M,S)-Nets
Niederreiter (1987), generalizing an idea of Sobol’ (1967)
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Example
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Sampling Evenly
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Sampling Evenly
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Sampling Evenly
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Sampling Evenly
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Sampling Evenly
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Sampling Evenly
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Sampling Evenly
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Sampling Evenly
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Sampling Evenly
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Sampling Evenly
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Sampling Evenly
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Sampling Evenly
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Using Latin Squares
Two MOLS(3) yield an orthogonal array of strength two
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Latin Squares to (0,2,2)-net
Replace alphabet by {0,1,…,b-1} (here, base b=3)
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Latin Squares to (0,2,2)-net
Insert decimal points to obtain a (0,2,2)-net in base 3
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The Resulting (T,M,S)-Net
(0,2,2)-net in base 3
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Su Doku!
Now fill in with cosets of the linear code
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Vienna, Austria 1980sMadison, Wisconsin 1995
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Generalized Orthogonal Arrays
Mark Lawrence, Chief Risk Officer, Australia and New Zealand Banking Group
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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
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Vienna, Austria 1980sSalzburg, Austria 1995
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Ordered Orthogonal Arrays
Wolfgang Ch. Schmid and Gary Mullen
Introduced OOA concept Proved equivalence to (T,M,S)-nets constructions bounds
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OOA
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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
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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
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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
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Proof Idea
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Vienna, Austria 1980sSingapore 1995
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Nets from Algebraic Curves
Harald Niederreiter and Chaoping Xing ( here pictured with Sang Lin)
Global function fields with many rational places
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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
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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 )
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Example
First 5 points (constant polys)
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Example
First 10 pts (constant &linear)
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Example
First 15 points (constant & linear)
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Example
First 20 points (constant & linear)
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Example
First 25 points (all const & lin)
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Example
First 50 points
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Example
First 75 points
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Example
First 100 points
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Example – a (0,3,3)-net in base 5
All 125 points
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Example – a (0,3,3)-net in base 5
All 125 points – another viewpoint
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Vienna, Austria 1980s
Heidelberg, Germany 1995
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Vienna, Austria 1980sHoughton, Michigan 1995
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From Codes to Nets
Yves Edel and Juergen Bierbrauer
Digital nets from BCH codes . . . and twisted BCH codes
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Vienna, Austria 1980sMoscow, Russia 1995
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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
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Vienna, Austria 1980sAuburn Alabama 1995
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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?”
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Vienna, Austria 1980sLaramie, Wyoming 1996
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Vienna, Austria 1980sOutside Laramie
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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
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Vienna, Austria 1980sWinnipeg, Manitoba 1997
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Vienna, Austria 1980sWinnipeg, March 1997
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Vienna, Austria 1980s
University of Manitoba
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Vienna, Austria 1980s
University of Nebraska
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Generalized Rao Bound
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Ordered Hamming Scheme
Doug Stinson and WJM
Self-dual association scheme generalising the Hamming schemes Duality between codes and OOAs MacWilliams identities, LP bound
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Ordered Hamming Scheme
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Ordered Hamming Scheme
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How to Learn of New Results
Vladimir Levenshtein BCC at Queen Mary & Westfield College (qmul)“Look at this paper by Rosenbloom and Tsfasman”
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RT Codes
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RT Codes
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Dual Codes and MacWilliams
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Dual Codes and MacWilliams
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MacWilliams Identity (Stinson/WJM)
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Duality: RT codes and OOAs
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St. Petersburg, Russia 1999
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Uniform Distributions
Steven Dougherty and Maxim Skriganov
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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
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Houghton, Michigan
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Vienna, Austria 1980sWinnipeg, Manitoba 1997
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The Dual Plotkin Bound
Terry Visentin and WJM
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Vienna, Austria 1980sSalzburg, Austria 1995
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The State of the Art
Wolfgang Ch. Schmid and Rudi SchurerMany contributionsBut also a comprehensive on-line table of parameters with links to literature
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Thank You
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This is the text I wantBUT THIS IS BETTERNOW WE HAVE ANOTHER OPTIONVIENNA, AUSTRIA: MAY 1986
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