NX-sequences site


This site is subject to change. Please revisit to keep updated.
Coming soon (April 2003): kash script for Niederreiter-\"Ozbudak sequences,
according bibliography update
Last updates:
2003-04-09 Bugfix (matters of format, not content) in libseq-formatted matrices fkmat511/63 (Dank an R.Schuerer),
2002-08-26, updated nxs??m?? and their t-values (Dank an A.Keller )
2002-05-22, niedxing_v3.kash: improved comments, msdos/win capable (?), b > 2 implemented, but not working correctly -_- (yet:)
2002-03-07, bugfix re 0/1 transposition in fkmats
2001-08-17, added t-value table for matrices up to s=32)
Maintained by Gottlieb.Pirsic@oeaw.ac.at

Papers

Original papers by Niederreiter and Xing

Preprints:

Generator matrices of NX-sequences

Name convention: nxs..m..

nx - signifies Niederreiter/Xing construction
s.. - dimension = no. of matrices
m.. - log no. of points, precision, size of matrices

File format:

First entry : m
Second entry : s
Next s*m entries : row vectors of the first, second,..., s-th matrix, encoded as decimal number. The vector consists of the digits of the b-ary expansion of the number, most significant bits corresponding to the initial coordinates of the vector. (E.g. b = 2, m = 8: entry 11 <-> row vector [0 0 0 0 1 0 1 1] )

All of the following matrices are constructed in base b = 2.



nxs04m42, nxs05m42, nxs06m42, nxs07m42, nxs08m42, nxs09m32,
nxs10m32, nxs11m32, nxs12m32, nxs13m32, nxs14m32, nxs15m32,
nxs16m32, nxs17m32, nxs18m32, nxs19m32, nxs20m32, nxs21m32,
nxs22m32, nxs23m32, nxs24m32, nxs25m32, nxs26m32, nxs27m32,
nxs28m32, nxs29m32, nxs30m32, nxs31m32, nxs32m32

All files in one .tgz archive : nxsmats.tgz
For t-values look here . Each column is one dimension, m increases with the row number (m grows down, s across). Also a transposed version is available as well as one transposed (i.e. each row is one dimension) and with propagation rules applied.

( For non-specialists: the t-values can be interpreted e.g. as an exponential factor in the error bound of numerical integration.)

Sources, Programs, Links

The above format is designed for calculating the t-value. For calculating the actual points, the file under this link is a tar-gzip compressed collection of the above matrices for use with the libseq-library of Ilja Friedel and Alexander Keller.
(ATTENTION: As Thomas Kollig thankfully pointed out, there had been an error in the libseq-formatted files provided earlier: the intended original matrices would have been like the entries in the files but with 0 and 1 interchanged.)
Under this link you can get an example C++ program (MS Windows version here) implementing the generation of the points (not the matrices!) (tested under Linux (g++) and Windows (cygwin) ).

Two further HIGHDIMENSIONAL matrices in the libseq format:
base 16, dimension 63, matrix size 30, t at most 6 : fkmat63b16
base 64, dimension 511, matrix size 32, t at most 28 : fkmat511b64
(Note: the TABLE_LENGTH in `options.h' of `libseq' may have to be increased to use this matrices!)

Please contact me at Gottlieb.Pirsic@oeaw.ac.at if you need larger matrix sizes ( = more points = higher precision ) for any of the above dimensions/bases or any further clarification.


A KASH script for generation of the matrices of the base 2 sequences in the above dimensions is available under this link. (Older version: here) This file (prototypes) is for variable declaration and this file (ffdata) holds function field data - here is where you may add further high rational place count function fields. All three files (niedxing_v?.kash, prototypes, ffdata) are required to exist in the same directory (e.g. your KASH directory) for the script to run correctly.
After the script is loaded into the KASH system, type

calc_c(m,s);
replacing m and s with your desired parameters. The matrices are stored in the variable c, an s by m by m array, and output into the file "matout" in the above format.

A C++ program is available on demand, for s at most 16. (However, I still don't trust it fully yet, so preferrably please use the above KASH script... )


Numbers, further data

Have a look at some numerical experiment results.
An (uncommented, undocumented) implementation of chapter 5.6 ( about explicit towers of algebraic function fields) of the book of Niederreiter and Xing, "Rational Points on Curves over Finite Fields" can be found here.

Pictures

Here is a nice 3D interactive stereo scatter plot of a Niederreiter-Xing sequence (needs JavaScript capabilities).

Almost unrelated (but pretty): a function plot (ca. 2M color postscript file) of (w-1)^4 = (v^3+1) over the complex plane, where x+i*y=v, z=abs(w) and color is arg(w). Observe three visible ramification points with different exponent d=3 (fourth is at infinity) and four leaves, therefore by the Hurwitz genus formula this is a Riemann surface topologically equivalent to a sphere with three handles (g=3). There is a triple zero of w at v=0.

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