A New Class of Linear Codes

Abstract

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.