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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