Sequences of linear codes where the rate times distance grows rapidly

TL;DR

This paper introduces new constructions of sequences of linear codes over arbitrary fields with rapidly growing product of rate and minimum distance, surpassing previous bounds and demonstrating significant potential for coding efficiency.

Contribution

The paper presents a novel recursive construction of linear codes with rapidly increasing rate-distance product, and provides examples including Reed-Muller subsequences with asymptotic growth.

Findings

Constructed sequences with $k_id_i/n_i > ext{growing function}$

Demonstrated rapid growth of $k_id_i/n_i$ in new code families

Provided asymptotic analysis for Reed-Muller subsequences

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 $c=\log_2(3/2)\approx 0.585$.