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 $k_i{d}_i/n_i$ grows quickly in the sense that $k_i{d}_i/n_i>\sqrt{k_i}-1>2i-1$. 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 $c={\log }_2(3/2)\approx 0.585$.

13 pages