Large Sets of Orthogonal Sequences Suitable for Applications in CDMA Systems

Abstract

In this paper, we employ the so-called semi-bent functions to achieve significant improvements over currently known methods regarding the number of orthogonal sequences per cell that can be assigned to a regular tessellation of hexagonal cells, typical for certain code-division multiple-access (CDMA) systems. Our initial design method generates a large family of orthogonal sets of sequences derived from vectorial semibent functions. A modification of the original approach is proposed to avoid a hard combinatorial problem of allocating several such orthogonal sets to a single cell of a regular hexagonal network, while preserving the orthogonality to adjacent cells. This modification increases the number of users per cell by starting from shorter codewords and then extending the length of these codewords to the desired length. The specification and assignment of these orthogonal sets to a regular tessellation of hexagonal cells have been solved regardless of the parity and size of m (where 2 m is the length of the codewords). In particular, when the re-use distance is D = 4 the number of users per cell is 2 m−2 for almost all m, which is twice as many as can be obtained by the best known methods.

Three classes of balanced vectorial semi-bent functions

“…Due to their almost optimal nonlinearity, semi-bent functions can resist linear approximation attacks [25,36] and fast correlation attacks [37] (assuming they possess a suitable resiliency order) when used in symmetric ciphers. Furthermore, semi-bent functions can be used to construct orthogonal variable spreading factor codes which are suitable for applications in synchronous CDMA communication systems [31,48,49].…”

“…Johansson and Pasalic [32] constructed the MM class of balanced correlation immune plateaued functions, which can be vectorial semi-bent functions under certain conditions. Zhang et al [49] showed that (n, n/2 + 1) vectorial semi-bent functions (stemming from the MM class) can be used to construct the orthogonal sequence sets suitable for applications in CDMA systems.…”

Construction and assignment of orthogonal sequences and zero correlation zone sequences for applications in CDMA systems

“…In addition, the cross-correlation of the sequences in a given cell between non-neighboring cells should be sufficiently small and lie in the range [2 m/2 , 2 (m+2)/2 ]. A usual way of constructing spreading codes in these systems is to use correlation-constrained sets of Hadamard matrices [7,8,10,12,13].…”

“…The so-called re-use distance D reflects the ability to use the same codewords in non-adjacent cells that are at distance D from the cell where these codewords have originally been placed. The re-use distance is discussed in detail in [8,13].…”

Designing plateaued Boolean functions in spectral domain and their classification

“…This class of functions has a wide range of applications such as cryptography, in the design of sequences for communications, and related combinatorics and designs. For instance, vectorial plateaued functions were efficiently used to provide large sets of orthogonal binary sequences suitable for CDMA (Code Division Multiple Access) applications which resulted in a significant improvement of the number of users per cell in regular hexagonal networks [25]. The design methods of plateaued functions have been addressed in several works in the past twenty years [2,5,22,16,9,19,23,27], but none of these construction methods is generic and furthermore a simple characterization of these functions in the Walsh domain has not been utilized as a construction rationale.…”