论文标题
关于使用双代码的代码字解码
On Decoding Using Codewords of the Dual Code
论文作者
论文摘要
我们提出了新颖的解码方案，用于使用双代码的最小重量编码器对块代码进行硬和软决策解码。将描述用于使用多项式的循环代码的解码方案，但是，可以对非循环代码进行修改，并且可以直截了当。艰难的决策解码计算综合征多项式，这是接收到多项式与双重编码的乘积。获得这些综合征多项式的正确循环移位，并将非零位置对这些移位进行计数。该计数的值是一个可靠性度量，可用于定位误差以及非错误位置。这种可靠性度量是艰苦决策解码算法的各种变体的基础。即使频道中没有可靠性信息，也可以使用迭代误差的解码方案以及使用措施固有的可靠性信息进行解码。此外，我们将展示如何包括来自渠道的可靠性信息，以获取软决策解码方案。我们得出了位，相信繁殖和对新方案的多数逻辑之间的关系。作为说明功能的示例，我们使用BCH和Reed-Muller代码作为二进制代码的示例，以及非二进制代码的RS代码。除了Plotkin结构外，我们还记得一个已知的结果，即被一个位置刺穿的芦苇毛刺代码是循环的，因此相当于BCH代码的特殊情况。将给出一些示例的硬性和软决策解码的仿真结果，并与文献的结果进行比较。最后，我们分析了Plotkin结构的软决策解码，并得出了两个代码之一使用$ 3 $ dB更好的通道（也称为通道极化）。
We present novel decoding schemes for hard and soft decision decoding of block codes using the minimal weight codewords of the dual code. The decoding schemes will be described for cyclic codes where polynomials can be used, however, the modification for non-cyclic codes is possible and straight forward. The hard decision decoding calculates syndrome polynomials which are the product of the received polynomial with dual codewords. Proper cyclic shifts of these syndrome polynomials are obtained and the non-zero positions are counted componentwise for these shifts. The values of this counting are a reliability measure and can be used for locating the error and also the non-error positions. This reliability measure is the basis for various variants of hard decision decoding algorithms. Decoding schemes with iterative error reduction are possible as well as information set decoding using the inherent reliability information of the measure even if there is no reliability information from the channel. Further, we will show how reliability information from the channel can be included in order to obtain soft decision decoding schemes. We derive the relation between bit flipping, believe propagation, and majority logic decoding to the novel schemes. As examples to illustrate the functioning we use BCH and Reed-Muller codes as examples for binary codes, and RS codes for non-binary codes. Besides the Plotkin construction we recall a known result that Reed-Muller codes punctured by one position are cyclic and thus, are equivalent to special cases of BCH codes. Simulation results for hard and soft decision decoding will be given for several examples and compared with results from literature. Finally, we analyze the soft decision decoding of the Plotkin construction and derive that one of the two codes uses a $3$ dB better channel (also known as channel polarization).
