Performance and analysis of Quadratic Residue Codes of lengths less than 100

In this paper, the performance of quadratic residue (QR) codes of lengths within 100 is given and analyzed when the hard decoding, soft decoding, and linear programming decoding algorithms are utilized. We develop a simple method to estimate the soft decoding performance, which avoids extensive simulations. Also, a simulation-based algorithm is proposed to obtain the maximum likelihood decoding performance of QR codes of lengths within 100. Moreover, four important theorems are proposed to predict the performance of the hard decoding and the maximum-likelihood decoding in which they can explore some internal properties of QR codes. It is shown that such four theorems can be applied to the QR codes with lengths less than 100 for predicting the decoding performance. In contrast, they can be straightforwardly generalized to longer QR codes. The result is never seen in the literature, to our knowledge. Simulation results show that the estimated hard decoding performance is very accurate in the whole signal-to-noise ratio (SNR) regimes, whereas the derived upper bounds of the maximum likelihood decoding are only tight for moderate to high SNR regions. For each of the considered QR codes, the soft decoding is approximately 1.5 dB better than the hard decoding. By using powerful redundant paritycheck cuts, the linear programming-based decoding algorithm, i.e., the ACG-ALP decoding algorithm performs very well for any QR code. Sometimes, it is even superior to the Chase-based soft decoding algorithm significantly, and hence is only a few tenths of dB away from the maximum likelihood decoding. Yong Li, Qianbin Chen and Hongqin Liu are with the Key Lab of Mobile Communication, Chongqing university of Posts and Telecommunications, Chongqing, 400065 China (e-mail: {yongli, hongqingliu, chenqb}@cqupt.edu.cn). Trieu-Kien Truong is with the Department of Information Engineering, I-Shou University, Kaohsiung Country 84001, Taiwan (e-mail: [email protected]), and is also with the Department of Computer Science and Engineering, National Sun Yat-sen University, Taiwan. August 26, 2014 DRAFT