$\texttt{GF}(p^{n^{2}})$ 中的多速率信道编码
Multiple-Rate Channel Codes in $\texttt{GF}(p^{n^{2}})$
AI总结：
本文提出一种从复合域现有信道编码推导多速率编码的方案，利用基域上的子编码实现固定分组长度下任意长度消息的编解码，支持多种信息速率，并配备基于正交投影算子的解码策略。
摘要：
在 $\texttt{GF}(q^{n})$ 上定义的编码 $\mathcal{C}(n, k, d)$ 通常被设计为将 $k$ 符号的用户数据编码为长度为 $n$ 的码字，从而实现固定速率编码。本文提出了一种编码程序，从复合域 $\texttt{GF}(q^{n})$ 上定义的现有信道编码中推导出多速率编码。形式上，通过将 $\texttt{GF}(q^{n})$ 的一个符号视为基域 $\texttt{GF}(q)$ 上的 $n$ 元组，所提编码方案采用定义在 $\texttt{GF}(q)$ 上的子编码 $\mathcal{C}_{1}(n, 1), \mathcal{C}_{2}(n, 2), \ldots, \mathcal{C}_{n}(n, n)$ 来编码任意长度的用户消息，并具备可变速率特性。随后，与传统的长度为 $n$ 的分组编码不同，推导出的固定分组长度为 $n$（在 $\texttt{GF}(q^{n})$ 上）的多速率编码可用于编码和解码任意长度 $|{\bf m}| = k, k+1, \ldots, kn$ 的用户消息 ${\bf m}$（在 $\texttt{GF}(q)$ 上），从而支持一系列信息速率——包括编码速率 $1/n, 2/n, \ldots, (k-1)/n$，以及现有的编码速率 $k/n$。所提的多速率编码方案还配备了解码策略，其中通过使用正交投影算子的简单程序来识别变长子编码用户消息。
英文摘要：
A code $\mathcal{C}(n, k, d)$ defined over $\texttt{GF}(q^{n})$ is conventionally designed to encode a $k$-symbol user data into a codeword of length $n$, resulting in a fixed-rate coding. This paper proposes a coding procedure to derive a multiple-rate code from existing channel codes defined over a composite field $\texttt{GF}(q^{n})$. Formally, by viewing a symbol of $\texttt{GF}(q^{n})$ as an $n$-tuple over the base field $\texttt{GF}(q)$, the proposed coding scheme employs children codes $\mathcal{C}_{1}(n, 1), \mathcal{C}_{2}(n, 2), \ldots, \mathcal{C}_{n}(n, n)$ defined over $\texttt{GF}(q)$ to encode user messages of arbitrary lengths and incorporates a variable-rate feature. In sequel, unlike the conventional block codes of length $n$, the derived multiple-rate code of fixed blocklength $n$ (over $\texttt{GF}(q^{n})$) can be used to encode and decode user messages ${\bf m}$ (over $\texttt{GF}(q)$) of arbitrary lengths $|{\bf m}| = k, k+1, \ldots, kn$, thereby supporting a range of information rates - inclusive of the code rates $1/n, 2/n, \ldots, (k-1)/n$, in addition to the existing code rate $k/n$. The proposed multiple-rate coding scheme is also equipped with a decoding strategy, wherein the identification of children encoded user messages of variable length are carried out through a simple procedure using {\it orthogonal projectors}.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
