A New Class of Linear Codes

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 gives a family of codes of length $n$, asymptotic distance $1/2$ and rate $\Omega(n^{-\varepsilon})$ for any fixed small $\varepsilon>0$, improving our initial construction. Such rate is also asymptotically better than the one by Kschischang and Tasbihi obtained by concatenating a Reed--Solomon with Reed--Muller, improving by a factor in $\Omega(n^{1/2}/\log(n))$.