Asymptotic estimation of error fraction corrected by binary LDPC code

Abstract

This paper considers new lower bound on fraction of guaranteed corrected errors while decoding the same binary low-density parity-check (LDPC) codes with constituent single parity-check (SPC) and Hamming codes using the same iterative low-complex hard-decision algorithm as in previous works of V. Zyablov and M. Pinsker in 1975 and V. Zyablov, R. Johannesson and M. Loncar in 2009. The number of decoding iterations, required to correct the errors, is a logarithmic function of the code length. The fraction of guaranteed correctable errors computed numerically for various choices of LDPC code parameters with constituent SPC and Hamming codes shows that proposed lower bound gives the better results than previously known best lower bounds obtained by V. Zyablov and M. Pinsker in 1975 for Gallager's LDPC codes and A. Barg and A. Mazumrad for Hamming code-based LDPC (H-LDPC) codes in 2011. Some of obtained numerical results are represented at the end of the paper to demonstrate these improvements.

The error-correcting capabilities of low-complexity decoded H-LDPC code as irregular LDPC code

“…In this paper for the first time we obtained the lower-bound on the guaranteed corrected errors fraction for H-LDPC code under the second decoding algorithm. Numerical results, represented at the end of the paper, for the lower-bound, obtained in this paper for H-LDPC code under the second decoding algorithm, significantly exceed the numerical results for the best known lower-bounds, obtained previously in [12], [13] for H-LDPC code under the first decoding algorithm.…”

“…Let us consider the first decoding algorithm A, proposed in [12], [13]. This algorithm uses the properties of the constituent Hamming code.…”

“…For H-LDPC code under the decoding algorithm A the lower-bound on fraction of the guaranteed corrected errors was obtained in [12], [13]. Let α 0 ω γ denote the fraction of the guaranteed corrected errors obtained in [13].…”

“…Then in [12], [13] a new lower bound for generalized regular LDPC codes was obtained. It was shown that numerical results of the lower bound [12], [13] for H-LDPC codes exceed similar results of the previously known best lower-bounds on the error correcting capabilities of H-LDPC codes [11].…”

“…The first decoding algorithm uses the properties of the constituent Hamming code. The lower-bound on the guaranteed corrected errors fraction for the H-LDPC codes under this decoding algorithm was obtained in [12], [13]. The second decoding algorithm considers H-LDPC as the irregular LDPC code and uses well-known majority decoding algorithm.…”