Sequences of linear codes where the rate times distance grows rapidly

Abstract

For a linear code $C$ of length $n$ with dimension $k$ and minimum distance $d$, it is desirable that the quantity $kd/n$ is large. Given an arbitrary field $\mathbb{F}$, we introduce a novel, but elementary, construction that produces a recursively defined sequence of $\mathbb{F}$-linear codes $C_1,C_2, C_3, \dots$ with parameters $[n_i, k_i, d_i]$ such that $k_id_i/n_i$ grows quickly in the sense that $k_id_i/n_i>\sqrt{k_i}-1>2i-1$. Another example of quick growth comes from a certain subsequence of Reed-Muller codes. Here the field is $\mathbb{F}=\mathbb{F}_2$ and $k_i d_i/n_i$ is asymptotic to $3n_i^{c}/\sqrt{\pi\log_2(n_i)}$ where $c=\log_2(3/2)\approx 0.585$.