Sequences of linear codes where the rate times distance grows rapidly

10/04/2021

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by   Faezeh Alizadeh, et al.

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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_id_i/n_i grows quickly in the sense that k_id_i/n_i>√(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/√(πlog_2(n_i)) where c=log_2(3/2)≈ 0.585.