Results on punctured LDPC codes

In this paper we study some fundamental properties of punctured LDPC codes. We first prove that for any ensemble of LDPC codes, there exists a puncturing threshold p*. We then find lower bounds on the achievable rates of punctured codes over general MBIOS channels. These bounds are satisfied by using only one encoder and decoder for all rates. We then prove that for any rates R₁ and R₂ satisfying 0 < R₁ < R₂ < 1, there exists an ensemble of LDPC codes with the following property. The ensemble can be punctured from rate R₁ to R₂ resulting in asymptotically good codes for all rates R₁ ≤ R ≤ R₂. Specifically, this implies that rates arbitrarily close to one are achievable via puncturing. We also show that punctured LDPC codes are as good as ordinary LDPC codes. For binary erasure channel (BEC) and arbitrary positive numbers R₁ < R₂ < 1, we prove the existence of the sequences of punctured LDPC codes that are capacity achieving for all rates R₁ ≤ R ≤ R₂. Based on the above observations, we then propose a method to design good punctured LDPC codes over a broad range of rates. The method is very simple and does not suffer from the performance degradation at high rates. Finally, we show that punctured codes might be useful for proof of the existence of capacity-achieving LDPC codes over memoryless binary-input output-symmetric channels.