Linear complexity problems of level sequences of Euler quotients and their related binary sequences

The Euler quotient modulo an odd-prime power $p^r~(r>1)$ can be uniquely decomposed as a $p$-adic number of the form $$ \frac{u^{{(p-1)p{r-1}}} -1}{p^r}\equiv a0(u)+a1(u)p+\ldots+a{r-1}(u)p^{r-1} \pmod {p^r},~ \gcd(u,p)=1, $$ where $0\le aj(u)<p$ for $0\le j\le r-1$ and we set all $a_j(u)=0$ if $\gcd(u,p)>1$. We firstly study certain arithmetic properties of the level sequences $(aj(u)){u\ge 0}$ over $\mathbb{F}p$ via introducing a new quotient. Then we determine the exact values of linear complexity of $(aj(u)){u\ge 0}$ and values of $k$-error linear complexity for binary sequences defined by $(aj(u))_{u\ge 0}$.