Narrow Results

Boolean functions are fundamental bricks in the development of various applications in Cryptography and Coding theory by making benefit from the weights of related Boolean functions (Walsh spectrum). Towards this, the discrete Fourier transform (Walsh–Hadamard) plays a pivotal tool. The work in this paper is dedicated towards the algebraic and numerical degrees, together with the relationship between weights of Boolean function and their Walsh transforms. We introduce Walsh matrices and generalize them to any arbitrary Boolean function. This improves the complexity in computation of Walsh–Hadamard and Fourier transform in certain cases. We also discuss some useful results related to the degree of the algebraic normal form using Walsh–Hadamard transform.

Trajectory representation model has been proposed to describe non-rigid deformation. An optimal trajectory space finding algorithm for 3D non-rigid structure from motion (OTSF-NRSFM) based on this model also has been proposed. However, the influence of the wavelet basis selection on the OTSF-NRSFM algorithm has still not been studied. To help OTSF-NRSFM researchers select wavelet basis properly, we investigated the influences of wavelet basis selection. Two typical wavelet bases, DCT basis and WHT basis, are discussed in this paper. The spectrum properties of wavelet basis and feature point trajectory, trajectory representation results on synthetic shark data, OTSF-NRSFM reconstruction results on synthetic data and real data are analyzed. The results show that the wavelet selection has much influence on OTSF-NRSFM reconstruction results of some non-rigid feature points, which have complicated trajectory. This paper gives researchers some inspiration about wavelet basis selection in OTSF-NRSFM algorithm.

The cross-correlation of two generalized Boolean functions defined on ${\mathbb{Z}}_2^n$ with values in ${\mathbb{Z}}_q$ is discussed in literature for $q=2^2$. We generalize this result for $q=2^s;$$s\ge 2$ and show that the existing result is a special case of our work. Further, we give some constructions of gbent functions.

We consider the problem of computing the weight distribution of a linear code of dimension k over a composite finite field ${\mathbb{F}}_q$ where q = 2^m. Due to the trace function we reduce the arithmetic operations over the composite field to those over a prime field. This allows us to apply a transform of Walsh-Hadamard type. The codes are represented by their characteristic vector with respect to a given generator matrix and a generator matrix of the k-dimensional simplex code $S_{q:k}$. The developed algorithm has complexity O(kmq^k−1).