= Solution
<Irreducible character degrees>[Irreducible degrees] divide the <order of a finite group>[order] of a finite $2$-group and satisfy
$$
\sum_i d_i^2=|G|=16.
$$
The number of <linear characters> is $|G/G'|$. For a <nontrivial group>[nontrivial] finite $2$-group this is at least $4$ unless the group is <abelian group>[abelian], and every <nonlinear irreducible character>[nonlinear] degree is at least $2$. A degree $4$ constituent is therefore impossible in the nonabelian case. If there are $\ell$ linear and $m$ degree-two characters, then
$$
\ell+4m=16,\qquad \ell\in\{4,8,16\}.
$$
Thus the only degree collections are
$$
\boxed{1^{16}},\qquad
\boxed{1^8,2^2},\qquad
\boxed{1^4,2^3},
$$
with respectively $r=16,10,7$.
For $1^{16}$ take the <cyclic group> $G=C_{16}=\langle g\rangle$. Every <conjugacy class> is a <singleton set>[singleton], and its <character table> is
$$
\chi_j(g^k)=\zeta_{16}^{\,jk},
\qquad 0\leq j,k<16,
$$
where $\zeta_{16}=e^{2\pi i/16}$.
For $1^8,2^2$, take $G=D_8\times C_2$, where
$D_8=\langle r,s:r^4=s^2=1,\ srs=r^{-1}\rangle$ and
$C_2=\langle z\rangle$. On the five classes
$$
1,\quad r^2,\quad\{r,r^3\},\quad
\{s,r^2s\},\quad\{rs,r^3s\},
$$
the four linear characters of $D_8$, indexed by
$\varepsilon,\delta\in\{\pm1\}$, and its degree-two character are
$$
\chi_{\varepsilon,\delta}=(1,1,\varepsilon,\delta,\varepsilon\delta),
\qquad
\psi=(2,-2,0,0,0).
$$
The ten classes of $G$ are $(C,1)$ and $(C,z)$. Its full table consists of
$$
(\chi\otimes\eta)(C,z^e)=\chi(C)\eta^e,
$$
where $\chi$ runs over the above five rows and
$\eta\in\{1,-1\}$ runs over the two characters of $C_2$. This explicitly gives eight linear and two degree-two rows.
For $1^4,2^3$, take the <dihedral group>
$$
D_{16}=\langle r,s:r^8=s^2=1,\ srs=r^{-1}\rangle.
$$
Order its seven classes as
$$
1,\ r^4,\ \{r,r^7\},\ \{r^2,r^6\},\
\{r^3,r^5\},\
\{r^{2j}s\}_{j=0}^3,\ \{r^{2j+1}s\}_{j=0}^3.
$$
The four linear rows are
$$
\chi_{\varepsilon,\delta}
=(1,1,\varepsilon,1,\varepsilon,\delta,\varepsilon\delta),
\qquad \varepsilon,\delta=\pm1.
$$
The three degree-two rows, for $k=1,2,3$, are
$$
\psi_k=
\left(
2,\ 2(-1)^k,\
2\cos\frac{k\pi}{4},\
2\cos\frac{k\pi}{2},\
2\cos\frac{3k\pi}{4},\
0,\ 0
\right).
$$
The row norms and mutual inner products, weighted by the displayed class sizes, verify irreducibility and completeness.
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