论文标题
伯尔曼代码：实现BEC能力的芦苇穆勒代码的概括
Berman Codes: A Generalization of Reed-Muller Codes that Achieve BEC Capacity
论文作者
论文摘要
我们确定一个二进制代码系列的结构类似于Reed-Muller（RM）代码，并将RM代码包括作为严格的子类。该家族中的代码表示为$ c_n（r，m）$，其双重表示为$ b_n（r，m）$。这些代码的长度为$ n^m $，其中$ n \ geq 2 $，$ r $是他们的“订单”。当$ n = 2 $时，$ c_n（r，m）$是订单$ r $和长度$ 2^m $的RM代码。这些代码的特殊情况与$ n $相对应的特殊情况是由Berman（1967）和Blackmore和Norton（2001）研究的。在Blackmore和Norton引入的术语之后，我们将$ B_N（R，M）$称为Berman Code，将$ C_N（R，M）$作为双Berman代码。我们使用递归皮肤的构造识别这些代码，并表明这些代码具有丰富的自动形态组，它们是由最小重量代码产生的，并且可以将其解码为最小距离的一半。使用Kumar等人的结果。 （2016年），我们表明这些代码在位图解码下实现了二元擦除通道（BEC）的能力。此外，除了双重传递性外，它们还满足了Reeves和Pfister所使用的所有代码属性，以表明RM代码达到了二进制输入无内存对称通道的能力。最后，当$ n $奇怪时，我们确定了包括$ b_n（r，m）$和$ c_n（r，m）$的大型Abelian代码，并实现了BEC容量。
We identify a family of binary codes whose structure is similar to Reed-Muller (RM) codes and which include RM codes as a strict subclass. The codes in this family are denoted as $C_n(r,m)$, and their duals are denoted as $B_n(r,m)$. The length of these codes is $n^m$, where $n \geq 2$, and $r$ is their `order'. When $n=2$, $C_n(r,m)$ is the RM code of order $r$ and length $2^m$. The special case of these codes corresponding to $n$ being an odd prime was studied by Berman (1967) and Blackmore and Norton (2001). Following the terminology introduced by Blackmore and Norton, we refer to $B_n(r,m)$ as the Berman code and $C_n(r,m)$ as the dual Berman code. We identify these codes using a recursive Plotkin-like construction, and we show that these codes have a rich automorphism group, they are generated by the minimum weight codewords, and that they can be decoded up to half the minimum distance efficiently. Using a result of Kumar et al. (2016), we show that these codes achieve the capacity of the binary erasure channel (BEC) under bit-MAP decoding. Furthermore, except double transitivity, they satisfy all the code properties used by Reeves and Pfister to show that RM codes achieve the capacity of binary-input memoryless symmetric channels. Finally, when $n$ is odd, we identify a large class of abelian codes that includes $B_n(r,m)$ and $C_n(r,m)$ and which achieves BEC capacity.
