Abstract
This paper deals with the specific construction of binary low-density parity-check (LDPC) codes. We derive lower bounds on the error exponents for these codes transmitted over the memoryless binary symmetric channel (BSC) for both the well-known maximum-likelihood (ML) and proposed low-complexity decoding algorithms. We prove the existence of such LDPC codes that the probability of erroneous decoding decreases exponentially with the growth of the code length while keeping coding rates below the corresponding channel capacity. We also show that an obtained error exponent lower bound under ML decoding almost coincide with the error exponents of good linear codes.
Highlights
Low-density parity-check (LDPC) codes [1] are known for their very efficient lowcomplexity decoding algorithms
This paper’s central question is: Are there LDPC codes that asymptotically achieve the capacity of binary-symmetric channel (BSC) under a lowcomplexity decoding algorithm? The following results help us construct LDPC code with specific construction and develop a decoding algorithm to answer yes to this question
Let there exist in the ensemble EG (`, n0, b0 ) of the G-LDPC codes a code with the code rate R2 that can correct any error pattern of weight up to bωt nc while decoding with the bit-flipping algorithm A M
Summary
Low-density parity-check (LDPC) codes [1] are known for their very efficient lowcomplexity decoding algorithms. We need to introduce to the construction of G-LDPC code some “good” codes that reduce the number of errors from the channel in such a way that the low-complexity majority decoding can correct the rest errors. To show that the proposed construction asymptotically achieves the capacity of BSC we consider the estimation on error-exponent under the proposed low-complexity decoding algorithm. It worth mention that papers [7,8] introduce expander codes achieving the BSC capacity under an iterative decoding algorithm with low complexity. The authors of [9,10] have derived the upper and lower bounds on the G-LDPC codes error exponent under the ML decoding assumption. We compare the obtained lower bounds on the error exponents under the low-complexity decoding and the ML decoding. We evaluate the error exponents numerically for different code parameters
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