Justesen code[1]

The first asymptotically good codes. Rate is \(R_m=k/n=K/2N\geq R\) and the relative minumum distance satisfy \(\delta_m=d_m/n\geq 0.11(1-2R)\), where \(K=\left\lceil 2NR\right\rceil\) for asymptotic rate \(0<R<1/2\); see pg. 311 of Ref. [2].

The code can be improved and extend the range of \(R\) from 0 to 1 by puncturing, i.e., by erasing \(s\) digits from each inner codeword. This yields a code of length \(n=(2m-s)N\) and rate \(R=mk/(2m-s)N\). The lower bound of the distance of the punctured code approaches \(d_m/n=(1-R/r)H^{-1}(1-r)\) as \(m\) goes to infinity, where \(r\) is the maximum of 1/2 and the solution to \(R=r^2/(1+\log(1-h^{-1}(1-r)))\), and \(h\) is the binary entropy function.