Linear feedback shift registers (LFSRs) and feedback with carry shift registers (FCSRs) can be used in cryptography, communications and circuit testings. In 2014, Gu and Klapper investigated the imbalance properties of FCSR sequences and gave two bounds for the one symbol and two consecutive symbol cases, which were then improved in 2015. In 2016, the imbalance properties of LFSR subsequences were studied and an upper bound was deduced. In this paper, we revisit the imbalance properties of the above sequences and improve upon their bounds.

Boolean functions play an important role in symmetric ciphers. Various cryptographic criteria such as nonlinearity, balancedness, correlation immunity and transparency order are closely related to the Walsh support of a Boolean function. However, we did not know more about the possible structure of Walsh supports of Boolean functions until Carlet and Mesnager conducted an in-depth study of Walsh supports for Boolean functions in 2005, proposing a class of n-variable balanced Boolean functions with Walsh support$\mathbb{F}_{2}^{n}\backslash \{0\}$, for$n\geq 10$. For$n\leq 6$, it can be verified using the computer that there is no Boolean function with the Walsh Support$\mathbb{F}_{2}^{n}\backslash \{0\}$. Nevertheless, for$n=7,8,9$, whether there is an n-variable Boolean function whose Walsh support is$\mathbb{F}_{2}^{n}\backslash \{0\}$has been an open problem for many years. In this paper, we aim to solve this problem. For$n=7$, based on the technique of concatenation, we obtain a 7-variable function with Walsh support$\mathbb{F}_{2}^{n}\backslash \{a\}$through finding two 6-variable Boolean functions whose Walsh spectra are equal or opposite numbers to each other at only one point. Then, using the linear transformation, we naturally get a 7 -variable balanced Boolean function with Walsh support$\mathbb{F}_{2}^{n}\backslash \{0\}$. For$n=8,9$, we construct mathematically a balanced Boolean function whose Walsh support is$\mathbb{F}_{2}^{n}\backslash \{0\}$respectively. Therefore, we solve the above open problem completely.