# 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 F, we introduce a novel, but elementary, construction that produces a recursively defined sequence of F-linear codes *C*_{1} , *C*_{2} , *C*_{3,} . . . 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 > 2*i* − 1. Another example of quick growth comes from a certain subsequence of Reed-Muller codes. Here the field is F = F_2 and *k*_i *d*_i /*n*_i is asymptotic to 3*n*_i^*c* / \sqrt{π log_2 (*n*_i )} where *c* = log_2 (3/2) ≈ 0.585.