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, concatenating with a Reed-Solomon code we get a family of codes of length $n$ and rate $n^{-1/(2n+2)-2\varepsilon/(n+1)}+O(n^{-1/(n+1)})$ and relative distance $1/2+O(n^{-\varepsilon})$. This shows that, for a fixed length, the rate of the concatenation suggested by Kschischang and Tasbihi (2024) of a Reed-Solomon and a Reed-Muller code can be made an order of magnitude smaller than a concatenation of a Reed-Solomon with a large dimensional Shadow code, while still keeping the regime of relative distance $1/2$. Finally, we show that the square of a Shadow code behaves like a random code and the Shadow code itself has a decoding algorithm, which suggest that such class of codes has the potential to be interesting for cryptographic applications.

翻译：令 $n$ 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，与所有已知的相对距离渐近于 $1/2$ 的线性及非线性码族（如 Delsarte-Goethals 码）相比，我们构造的二元码具有指数级规模。此外，通过与 Reed-Solomon 码级联，我们得到一族长度为 $n$、码率为 $n^{-1/(2n+2)-2\varepsilon/(n+1)}+O(n^{-1/(n+1)})$、相对距离为 $1/2+O(n^{-\varepsilon})$ 的码。这表明对于固定长度，由 Kschischang 和 Tasbihi（2024）提出的 Reed-Solomon 码与 Reed-Muller 码级联的码率，可比 Reed-Solomon 码与高维 Shadow 码的级联码率降低一个数量级，同时仍保持 $1/2$ 的相对距离区间。最后，我们证明 Shadow 码的平方的行为类似于随机码，且 Shadow 码本身具有译码算法，这表明该类码在密码学应用中具有潜在价值。