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 C₁ , C₂ , C_3, . . . with parameters [n__i, k__i, d__i ] such that k_id_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 F = F_2 and k_i d_i /n_i is asymptotic to 3n_i^c / \sqrt{π log_2 (n_i )} where c = log_2 (3/2) ≈ 0.585.