Good error-correcting codes based on very sparse matrices

Abstract

We study two families of error-correcting codes defined in terms of very sparse matrices. "MN" (MacKay-Neal) codes are recently invented, and "Gallager codes" were first investigated in 1962, but appear to have been largely forgotten, in spite of their excellent properties. The decoding of both codes can be tackled with a practical sum-product algorithm. We prove that these codes are "very good," in that sequences of codes exist which, when optimally decoded, achieve information rates up to the Shannon limit. This result holds not only for the binary-symmetric channel but also for any channel with symmetric stationary ergodic noise. We give experimental results for binary-symmetric channels and Gaussian channels demonstrating that practical performance substantially better than that of standard convolutional and concatenated codes can be achieved; indeed, the performance of Gallager codes is almost as close to the Shannon limit as that of turbo codes.

Lower Computation and Storage Complexity of QC-LDPC Codes in Rayleigh Fading Channel

“…The signal noise ratio is set to be 20 dB to attain a packet error ratio of 3 10 − against the value fractional bandwidth. The curves in Fig.3 show that newly obtained codes cater a gain of 2.1 dB compared with RS codes at the worst condition of partial band ρ =1.…”

“…Low-density parity-check codes [1] have acquired considerable attention due to its near-capacity error execution and powerful channel coding technique with an adequately long codeword length. The performance of LDPC codes has been investigated in [2,3], and are encountered to outperform turbo codes.…”

“…Low-density parity-check codes [1] have acquired considerable attention due to its near-capacity error execution and powerful channel coding technique with an adequately long codeword length. The performance of LDPC codes has been investigated in [2,3], and are encountered to outperform turbo codes.The potency of LDPC codes is outclass over the additive White Gaussian noise (AWGN) channel [4], where coherent detection employing phase shift keying (PSK) [5] can be carried out with carrier phase estimation. Precise phase tracking and high-quality estimation of channel state information (CSI) is required for the appropriate execution of coherent detection.…”

Construction of Structured Nonbinary Low-Density Parity-Check Codes

“…This equation can be partitioned into M checks associated with M rows of H. When the parity check equation is not satisfied, then the error correction decoding is applied by means of the iterative message passing algorithm [1].…”

“…Low-density parity-check (LDPC) codes, after their "rediscovery" in late 90's [1], have attracted great research attention due to their excellent error-correcting performance and highly parallel iterative decoding scheme. They have become the industry standard for error correction coding, adopted for instance in the ETSI Digital Video Broadcasting (DVB) and the IEEE WiMAX.…”