Simple Nonalgebraic Decoding Techniques for Group Codes

Hailin Xiao, Ouyang Shan, Wu Xie

Quantum communication is a growing interdisciplinary field which combines classical communications and quantum mechanics. Quantum error correction coding is one of the key techniques in quantum communication. Nearly all of the classical error correction coding schemes have been transplanted to the domain of quantum communication, and the quantum counterparts of classical error correction coding techniques have been found. Based on the classical turbo product codes (TPCs) which is one of the most outstanding schemes in classical coding region, a new structure of the CSS-type quantum convolutional codes (QCC) as stabilizer sub-code of the quantum turbo product codes (QTPC) is presented. Firstly, CSS-type QCC stabilizer generator is constructed with the help of group theory and the basic principle of stabilizer coders, and the corresponding networks are described. Secondly, the interleaved coded matrix of the QTPC is obtained by quantum permutation SWAP gate definition. Finally, the corresponding relation between the quantum trace distance of QTPC decoding and the distance of classical TPCs decoding is obtained, and the scheme of QTPCs coding and decoding is completed. The coding and decoding of QTPCs have a highly regular structure and a simple design idea, and the networks are easy to realize.

Kenny Barrese

Martin Bossert, F. Hergert

Shumei Song, Shu Lin, Khaled Abdel-Ghaffar +1 more

Iterative algorithms for decoding over erasure-burst channels have attracted much interest lately due to their simplicity. Although the performance of these techniques is typically good, it may not be optimal if the underlying matrix used for decoding is not properly chosen. In this paper, we construct matrices that lead to optimal performance when used with iterative decoding over erasure-burst channels. We also develop a simple decoding technique that makes use of these same matrices to decode any linear code over error-burst channels. For cyclic codes, this algorithm is optimal.

Beomkyu Shin, Seokbeom Hong, Hosung Park +2 more

In this paper, a new decoding scheme is proposed to improve the error correcting performance of low-density parity-check (LDPC) codes in high signal-to-noise ratio (SNR) region by using post-processing. It behaves as follows: First, a conventional LDPC decoding is applied to received LDPC codewords one by one. Then, we count the number of word errors in a predetermined number of decoded codewords. If there is no word error, nothing needs to be done and we can move to the next group of codewords with no delay. Otherwise, we perform a proper post-processing which pro-ducesa new soft-valued codeword (this will befully explained in the main body of this paper) and then apply the conventional LDPC decoding to it again to recover the unsuccessfully decoded codewords. For the proposed decoding scheme, we adopt a simple product code structure which contains LDPC codes and simple algebraic codes as its horizontal and vertical codes, respectively. The decoding capability of the proposed decoding scheme is defined and analyzed using the parity-check matrices of vertical codes and, especially, the combined-decodability is derived for the case of single parity-check (SPC) codes and Hamming codes used as vertical codes. It is also shown that the proposed decoding scheme achieves much better error correcting capability in high SNR region with little additional decoding complexity, compared with the conventional LDPC decoding scheme.

Fernando Hernando, Kristine Lally, Diego Ruano