Abstract
Recent research has established close links between digital nets and coding theory. In fact, the problem of constructing good digital nets can now be viewed as the problem of constructing good linear codes in metric spaces that are more general than Hamming spaces. In this paper we report on the fascinating connections between digital nets and linear codes. In particular, we describe the duality theory for digital nets, the asymptotics of digital-net parameters, and the new concept of cyclic digital nets.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bierbrauer, J., Edel, Y., Schmid, W.Ch.: Coding-theoretic constructions for (t, m, s)- nets and ordered orthogonal arrays. J. Combin. Designs 10, 403–418 (2002)
Blahut, R.E.: Theory and Practice of Error Control Codes. Addison-Wesley, Reading, MA (1983)
Clayman, A.T., Lawrence, K.M., Mullen, G.L., Niederreiter, H., Sloane, N.J.A.: Updated tables of parameters of (t, m, s)-nets. J. Combin. Designs 7 381–393 (1999)
Edel, Y., Bierbrauer, J.: Construction of digital nets from BCH-codes. In: Niederreiter, H., et al. (eds.) Monte Carlo and Quasi-Monte Carlo Methods 1996, Lecture Notes in Statistics, Vol. 127, pp. 221–231. Springer, New York (1998)
Edel, Y., Bierbrauer, J.: Families of ternary (t, m, s)-nets related to BCH-codes. Monatsh. Math. 132 99–103 (2001)
Faure, H.: Discrépance de suites associées à un systéme de numération (en dimension s). Acta Arith. 41 337–351 (1982)
Helleseth, T., Klove, T., Levenshtein, V.I.: Hypercubic 4- and 5-designs from doubleerror-correcting BCH codes. Designs Codes Cryptogr. 28 265–282 (2003)
Larcher, G.: Digital point sets: analysis and application. In: Hellekalek, P., Larcher, G. (eds.) Random and Quasi-Random Point Sets, Lecture Notes in Statistics, Vol. 138, pp. 167–222. Springer, New York (1998)
Lawrence, K.M., Mahalanabis, A., Mullen, G.L., Schmid, W.Ch.: Construction of digital (t, m, s)-nets from linear codes. In: Cohen, S., Niederreiter, H. (eds.) Finite Fields and Applications, London Math. Soc. Lecture Note Series, Vol. 233, pp. 189208. Cambridge University Press, Cambridge (1996)
Lidl, R., Niederreiter, H.: Introduction to Finite Fields and Their Applications, rev. ed. Cambridge University Press, Cambridge (1994)
Niederreiter, H.: Low-discrepancy point sets. Monatsh. Math. 102 155–167 (1986)
Niederreiter, H.: Point sets and sequences with small discrepancy. Monatsh. Math. 104 273–337 (1987)
Niederreiter, H.: A combinatorial problem for vector spaces over finite fields. Discrete Math. 96 221–228 (1991)
Niederreiter, H.: Random Number Generation and Quasi-Monte Carlo Methods. SIAM, Philadelphia (1992)
Niederreiter, H.: Constructions of (t, m, s)-nets. In: Niederreiter, H., Spanier, J. (eds.) Monte Carlo and Quasi-Monte Carlo Methods 1998, pp. 70–85. Springer, Berlin (2000)
Niederreiter, H., Özbudak, F.: Constructions of digital nets using global function fields. Acta Arith. 105 279–302 (2002)
Niederreiter, H., Ă–zbudak, F.: Matrix-product constructions of digital nets. Finite Fields Appl. (to appear)
Niederreiter, H. Pirsic, G.: Duality for digital nets and its applications. Acta Arith. 97 173–182 (2001)
Niederreiter, H., Pirsic, G.: A Kronecker product construction for digital nets. In: Fang, K.-T., Hickernell, F.J., Niederreiter, H. (eds.) Monte Carlo and Quasi-Monte Carlo Methods 2000, pp. 396–405. Springer, Berlin (2002)
Niederreiter, H., Xing, C.P.: Rational Points on Curves over Finite Fields: Theory and Applications. London Math. Soc. Lecture Note Series, Vol. 285. Cambridge University Press, Cambridge (2001)
Niederreiter, H., Xing, C.P.: A construction of digital nets with good asymptotic behavior. Technical Report, Temasek Laboratories, National University of Singapore (2001)
Niederreiter, H., Xing, C.P.: Constructions of digital nets. Acta Arith. 102 189–197 (2002)
Ă–zbudak, F.: Elements of prescribed order, prescribed traces and systems of rational functions over finite fields. Designs Codes Cryptogr. (to appear)
Rosenbloom, M.Yu., Tsfasman, M.A.: Codes for the rn-metric. Problems of Inform. Transmission 33 45–52 (1997)
Schmid, W.Ch., Wolf, R.: Bounds for digital nets and sequences. Acta Arith. 78, 377–399 (1997)
Skriganov, M.M.: Coding theory and uniform distributions (Russian). Algebra i Analiz 13, 191–239 (2001)
Xing, C.P., Niederreiter, H.: Digital nets, duality, and algebraic curves. In: Niederreiter, H. (ed.) Monte Carlo and Quasi-Monte Carlo Methods 2002. Springer Berlin(to appear)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2004 Springer Basel AG
About this paper
Cite this paper
Niederreiter, H. (2004). Digital Nets and Coding Theory. In: Feng, K., Niederreiter, H., Xing, C. (eds) Coding, Cryptography and Combinatorics. Progress in Computer Science and Applied Logic, vol 23. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-7865-4_17
Download citation
DOI: https://doi.org/10.1007/978-3-0348-7865-4_17
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-9602-3
Online ISBN: 978-3-0348-7865-4
eBook Packages: Springer Book Archive