Bill Martin Mathematical Sciences and Computer Science
Worcester Polytechnic Institute
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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 wont talk about WPI is looking for
graduate students and visiting faculty
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Quadrature rules Numerical simulation Global optimization
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Random Pseudo-random (should fool an observer) Quasi-Random:
entirely deterministic, but has some statistical properties that a
random set should have
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Random (Monte Carlo) Lattice rules Latin hypercube sampling
(T,M,S)-nets
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A set N of N points inside [0,1) s An interval E = [0,a 1
)x[0,a 2 )x... x[0,a s ) 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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J. KoksmaE. Hlawka
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For any given shape (d 1,d 2,...,d s ), the unit cube is
partitioned into b m elementary intervals of this shape, each being
a translate of every other.
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Vienna, Austria 1980s
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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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Niederreiter (1987), generalizing an idea of Sobol (1967)
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Two MOLS(3) yield an orthogonal array of strength two
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Replace alphabet by {0,1,,b-1} (here, base b=3)
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Insert decimal points to obtain a (0,2,2)-net in base 3
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(0,2,2)-net in base 3
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Now fill in with cosets of the linear code
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Vienna, Austria 1980s Madison, Wisconsin 1995
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Mark Lawrence, Chief Risk Officer, Australia and New Zealand
Banking Group
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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 q t of these) appears
equally often So the total number of rows is.b t where 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 1980s Salzburg, Austria 1995
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Wolfgang Ch. Schmid and Gary Mullen Introduced OOA concept
Proved equivalence to (T,M,S)-nets constructions bounds
There exists a (T,M,S)-net in base b If and only if there
exists an OOA t, s, l, v) where s=S t=l=M-T v=b = b T
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Vienna, Austria 1980s Singapore 1995
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Harald Niederreiter and Chaoping Xing ( here pictured with Sang
Lin) Global function fields with many rational places
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For simplicity, assume q is a prime Let S = { p 1, p 2,..., p s
} be a subset of F q (or PG(1,q) ) Fix k >= 0 and create one
point for each polynomial f(x) in F q [x] of degree k or less In
the i th coordinate position, take f(p i )/q + f (1) (p i )/q 2
+... + f (k) (p i )/q k+1 where f (j) denotes the j th derivative
of f
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To illustrate, lets take q = 5 k = 2 S = { 1, 2, 3} inside F 5
For example, the polynomial f(x) = 3 x 2 + 4 x has 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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First 5 points (constant polys)
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First 10 pts (constant &linear)
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First 15 points (constant & linear)
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First 20 points (constant & linear)
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First 25 points (all const & lin)
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First 50 points
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First 75 points
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First 100 points
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All 125 points
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All 125 points another viewpoint
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Vienna, Austria 1980s Heidelberg, Germany 1995
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Vienna, Austria 1980s Houghton, Michigan 1995
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Yves Edel and Juergen Bierbrauer Digital nets from BCH codes
... and twisted BCH codes
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Vienna, Austria 1980s Moscow, Russia 1995
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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 1980s
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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 1980s
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Michael Adams Completed dissertation at U. Wyoming under Bryan
Shader Poset metrics for codes New constructions of nets Convincing
argument that MacWilliams identities DONT exist
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Vienna, Austria 1980s Winnipeg, Manitoba 1997
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Vienna, Austria 1980s Winnipeg, 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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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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Vladimir Levenshtein BCC at Queen Mary & Westfield College
(qmul) Look at this paper by Rosenbloom and Tsfasman
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St. Petersburg, Russia 1999
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Steven Dougherty and Maxim Skriganov
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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 1980s Winnipeg, Manitoba 1997
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Terry Visentin and WJM
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Vienna, Austria 1980s Salzburg, Austria 1995
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Wolfgang Ch. Schmid and Rudi Schurer Many contributions But
also a comprehensive on-line table of parameters with links to
literature
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Thank You
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Sga12345677890- qwery xcbaABKFQWFIOQWUFO: EIVNS U N n
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TTTThis is the text I want BBBBUT THIS IS BETTER NNNNOW WE HAVE
ANOTHER OPTION VVVVIENNA, AUSTRIA: MAY 1986
