Optimal rate and maximum erasure probability LDPC codes in binary erasure channel

Abstract

In this paper, we present a novel way for solving the main problem of designing the capacity approaching irregular low-density parity-check (LDPC) code ensemble over binary erasure channel (BEC). The proposed method is much simpler, faster, accurate and practical than other methods. Our method does not use any relaxation or any approximate solution like previous works. Our method works and finds optimal answer for any given check node degree distribution. The proposed method was implemented and it works well in practice with polynomial time complexity. As a result, we represent some degree distributions that their rates are close to the capacity with maximum erasure probability and maximum code rate.

Overhead minimization of online codes

“…Without loss of generality, we consider that the length of each block is one. Erasure channel losses a fraction δ of the packets and it is well-known that a code with rate R 1 − δ can achieve the capacity [3][4][5][6][7][8]. Let us define like the original online code that each check block with degree i is given by the XOR of i message bits, which are selected randomly.…”

“…According to the recent and new advances in error control coding theory, especially with using the regular [1] and irregular [2][3][4][5][6][7][8] form of Low Density Parity Check (LDPC) codes, efficient error correction schemes can be found. By using the LDPC codes and their well-known decoders such as Belief Propagation (BP) algorithm, one can achieve the whole channel capacity of the Binary Erasure Channel (BEC) [9][10][11][12] and approach to a rate, which is very close to channel capacity of Binary Symmetric Channel (BSC) [10], or Additive White Gaussian Noise Channel (AWGNC) [13].…”

Optimal rate irregular low-density parity-check codes in binary erasure channel

“…In the fourth approach which is developed in this paper, using semi-definite programming (SDP) is the main idea. In this method, instead of using some samples of non-linear DE constraint like the pervious methods, the whole space is considered [21,22]. Whereas relaxation of some constraints in the optimisation problem leads to a sub-optimal solution, in our method which is based on an exact constraint with no relaxation, the solution of SDP problem would be certainly optimal.…”

A fast convergence density evolution algorithm for optimal rate LDPC codes in BEC

“…In this paper, we introduce a group of codes with optimal rate and fast convergence behavior in decreasing the erased probability in compared with those in [16][17]. We first review the convergence behavior of the code in BEC by DE formula.…”