Let $n$ be a prime power, $r$ be a prime with $r\mid n-1$, and $\varepsilon\in (0,1/2)$. Using the theory of multiplicative character sums and superelliptic curves, we construct new codes over $\mathbb F_r$ having length $n$, relative distance $(r-1)/r+O(n^{-\varepsilon})$ and rate $n^{-1/2-\varepsilon}$. When $r=2$, our binary codes have exponential size when compared to all previously known families of linear and non-linear codes with relative distance asymptotic to $1/2$, such as Delsarte--Goethals codes. Moreover, our codes are linear.

翻译：设 $n$ 为素数幂，$r$ 为满足 $r\mid n-1$ 的素数，且 $\varepsilon\in (0,1/2)$。利用乘法特征和与超椭圆曲线理论，我们在 $\mathbb F_r$ 上构造了长度为 $n$、相对距离为 $(r-1)/r+O(n^{-\varepsilon})$、码率为 $n^{-1/2-\varepsilon}$ 的新型码。当 $r=2$ 时，我们的二元码相较于所有先前已知的相对距离渐近于 $1/2$ 的线性码与非线性码族（例如 Delsarte--Goethals 码）具有指数级规模。此外，我们的码是线性码。