Monomial codes were recently equipped with partial order relations, fact that allowed researchers to discover structural properties and efficient algorithm for constructing polar codes. Here, we refine the existing order relations in the particular case of Binary Erasure Channel. The new order relation takes us closer to the ultimate order relation induced by the pointwise evaluation of the Bhattacharyya parameter of the synthetic channels. The best we can hope for is still a partial order relation. To overcome this issue we appeal to related technique from network theory. Reliability network theory was recently used in the context of polar coding and more generally in connection with decreasing monomial codes. In this article, we investigate how the concept of average reliability is applied for polar codes designed for the binary erasure channel. Instead of minimizing the error probability of the synthetic channels, for a particular value of the erasure parameter p, our codes minimize the average error probability of the synthetic channels. By means of basic network theory results we determine a closed formula for the average reliability of a particular synthetic channel, that recently gain the attention of researchers.

中文翻译：

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二进制擦除通道的单项式编码的Bhattacharyya参数：从点向可靠性到平均可靠性

单项式代码最近配备了偏序关系，这一事实使研究人员能够发现结构性质和用于构造极性代码的有效算法。在这里，我们在二进制擦除通道的特定情况下细化了现有的顺序关系。新的阶次关系使我们更接近于由合成通道的Bhattacharyya参数的逐点求值引起的最终阶次关系。我们可以期望的最好仍然是部分订单关系。为了克服这个问题，我们呼吁采用网络理论中的相关技术。可靠性网络理论最近在极性编码的背景下使用，并且更普遍地与减少的单项编码结合使用。在本文中，我们研究了平均可靠性的概念如何应用于为二进制擦除通道设计的极性代码。对于特定的擦除参数p值，不是最小化合成通道的错误概率，而是我们的代码将合成通道的平均错误概率最小化。借助基本的网络理论结果，我们为特定合成通道的平均可靠性确定了一个封闭的公式，这一公式最近引起了研究人员的注意。