Let $p\equiv 1(\mathrm{mod}4)$ be a prime, $m$ a positive integer, $\frac{\varphi (p^m)}{2}$ the multiplicative order of $2$ modulo $p^m$, and let $q=2^{\frac{\varphi (p^m)}{2}}$, where $\varphi (\cdot )$ is the Euler's function. In this paper, we construct two classes of linear codes over ${\mathbb{F}}_q$ and investigate their weight distributions. By calculating two classes of special exponential sums, the desired results are obtained.