Duality for digital sequences
The \((t,m,s)-\)nets and the \((t,s)-\)sequences are classes of objects with an exceptionally well distribution in the \(s-\)dimensional unit cube \([0,1)^{s}.\) They have numerous applications in many branches of science. There are many analytical algorithms for their construction. The duality theory for digital nets has been introduced by \textit{H. Niederreiter} and \textit{G. Pirsic} [Acta Arith. 97, 173--182 (2001; Zbl 0972.11066)]. In this paper the duality theory is extended from finite digital nets to infinite digital sequences. In the Introduction the common concepts of the construction of digital nets and sequences over finite fields based on using generating matrices is reminded. The duality theory for digital nets is briefly presented. In Section 2 the definition of digital nets and sequences over finite fields with respect to an arbitrary bijection which is based on using generating matrices is reminded. The dual space \({\mathcal N}_{m}^{\bot}\) of the mth row space \({\mathcal N}_{m}\) of digital sequence is introduced. In Section 3 the duality theory for digital nets is extended to duality theory for digital sequences. The notion of minimum distance is defined. The necessary and sufficient conditions for generating a digital \(({\mathbf T},s)-\)sequences and a strict digital \(({\mathbf T},s)-\)sequences in the terms of minimum distance are obtained. In Section 4 the fundamental properties, that show which sequences of the dual space define a digital sequence and how the digital sequence can be obtained from the sequence \(({\mathcal M}_{m})_{m \geq 1}\) of the dual space chain, are obtained. In Definition 3 the notion of dual space chain is introduced. Theorem 1 confirms that for a given generating matrices \(C_{1}, \dots, C_{s}\) of digital sequences the associated sequence \(({\mathcal N}_{m}^{\bot})_{m \geq 1}\) is a dual space chain. Theorem 2 shows that for a given sequence \(({\mathcal M}_{m})_{m \geq 1}\) of dual space chain one can construct generating matrices \(C_{1}, \dots, C_{s}\) of a digital sequence such that for all \(m \geq 1\) the mth row space \({\mathcal N}_{m}\) satisfies \({\mathcal N}_{m}^{\bot} = {\mathcal M}_{m}.\) Theorem 3 presents a necessary and sufficient condition that to the sequence \(({\mathcal M}_{m})_{m \geq 1}\) of dual space chain correspond generating matrices \(C_{1}, \dots, C_{s}.\) In Section 5 it is shown how the construction of digital sequences of \textit{H. Niederreiter} and \textit{C. Xing} [Finite Fields Appl. 2, No. 3, 241-273 (1996; Zbl 0893.11029)] can be obtained by using the duality theory for digital sequences. The main result here is Theorem 4 where the concrete conditions for obtaining digital sequences of Niederreiter and Xing are presented.
\((t, m, s)\)-nets and \((t, s)\)-sequences over finite fields
scientific article; zbMATH DE number 5611796