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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

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0 0 1 01 1 1 00 1 1 1 0 1 01 1 1 00 10 1 0 1 1 01 1 00 0 1 1 0

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0 0 0 0 0 11 0 11 1 1 1 0 10 1 1 1 1 10 0 11 0 1 1 1 01 0 00 1 0 0 1 11 1 10 0 1 1 0 01 1 01 0 0 0 1 0 1 1 0

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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

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