This paper proposes a new iterative gradient descent decoding method for real number parity codes. The proposed decoder, named Gradient Descent Symbol Update (GDSU), is used for a class of low-density parity-check (LDPC) real-number codes that can be defined with parity check matrices which are similar to those of the binary LDPC from communication standards such as WiFi (IEEE 802.11), 5G. These codes have a simple and efficient two stage encoding that is especially appealing for the real number field. The Gradient optimization based decoding has been a relatively simple and fast decoding technique for codes over finite fields. We show that the GDSU decoder outperforms the gradient descent bit-flipping (GDBF) decoder for rates $1/2$, $2/3$, and has similar decoding performance for the $3/4$ rate of the IEEE 802.11 codes standard.

翻译：本文提出了一种针对实数奇偶校验码的新型迭代梯度下降解码方法。所提出的解码器命名为梯度下降符号更新（GDSU），适用于一类可通过奇偶校验矩阵定义的低密度奇偶校验（LDPC）实数码，其校验矩阵与WiFi（IEEE 802.11）、5G等通信标准中二进制LDPC码的校验矩阵相似。这类编码具有简洁高效的两阶段编码结构，在实数域中尤为适用。基于梯度优化的解码技术一直是有限域编码中相对简单快速的解码方法。我们证明，对于IEEE 802.11编码标准中码率为$1/2$和$2/3$的情况，GDSU解码器的性能优于梯度下降比特翻转（GDBF）解码器；而在$3/4$码率下，两者具有相近的解码性能。