2021/10/04 by Faezeh Alizadeh, Alizadeh, Faezeh, S. P. Glasby +3
Computer Science · #Coding theory and cryptography #Error Correcting Code Techniques #Cellular Automata and Applications
paper · doi:10.48550/arxiv.2110.01277
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 \mathbbF, we introduce a novel, but elementary, construction that produces a recursively defined sequence of \mathbbF-linear codes C1,C2, C3, … with parameters [ni, ki, di] such that kidi/ni grows quickly in the sense that kidi/ni>√(ki)-1>2i-1. Another example of quick growth comes from a certain subsequence of Reed-Muller codes. Here the field is \mathbbF=\mathbbF2 and ki di/ni is asymptotic to 3nic/√(πlog2(ni)) where c=log2(3/2)≈ 0.585.