Some classes of resilient functions over Galois rings and their linear characteristics

Let $R=\text{GR}(q^l,p^l)=\{r_1,\ldots,r_{q^l}\}$ be a Galois ring. Let $A_n(R)$ be a set of all affine functions $g(\vec x)=a_0+a_1x_1+\ldots+a_nx_n=a_0+\langle \vec a,\vec x \rangle$, where $\vec x=(x_1,\ldots,x_n)$, $a_0\in R$, $\vec a=(a_1,\ldots,a_n)\in R^n$. We present some classes of resilient function $f:R^n\to R$ with the specified value of linear characteristic $C(f)$, where $C(f)=\max_{a\in R\setminus \{0\}}\max_{g\in A_n(R)}\left|\sum\limits_{x_1,\ldots,x_n\in R}{\chi(af(\vec x)-g(\vec x))} \right|$ and $\chi$ is the canonical additive character of the ring $R$. In the paper, we describe the function $f$ using a branching construction of the given functions $f_1,\ldots,f_{r_{q^l}}$ in $n-1$ variables. We prove that in the case when the functions $f_1,\ldots,f_{r_{q^l}}$ are $k$-resilient, the resulting function $f$ is also $k$-resilient. Moreover, $C(f)\le C(f_{r_1})+\ldots+C(f_{r_{q^l}})$. We also describe the function $f(\vec x,\vec y)=\langle \phi(\vec x),\vec y\rangle+h(\vec x)$, where $n=2k$, $\phi:R^k\to R^{k}$, $h:R^k\to R$, $\vec x$, $\vec y\in R^k$. It is known that in the case $\phi(R^k)\subset (R^*)^k$ ($R^*$ is the group of units in the ring $R$) the function $f$ is $(k-1)$-resilient. We prove that in the case $|\phi^{-1}(\vec c)|\le t$ for all $\vec c\in R^k$ the enequality $C(f)\le q^{k(2l-1)}$ is true.