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}.

翻译：传统上，定义在 $\texttt{GF}(q^{n})$ 上的码 $\mathcal{C}(n, k, d)$ 旨在将 $k$ 个符号的用户数据编码为长度为 $n$ 的码字，从而实现固定速率的编码。本文提出了一种编码方法，用于从定义在复合域 $\texttt{GF}(q^{n})$ 上的现有信道码中导出多速率码。具体而言，通过将 $\texttt{GF}(q^{n})$ 中的一个符号视为基域 $\texttt{GF}(q)$ 上的一个 $n$ 元组，所提出的编码方案采用定义在 $\texttt{GF}(q)$ 上的子码 $\mathcal{C}_{1}(n, 1), \mathcal{C}_{2}(n, 2), \ldots, \mathcal{C}_{n}(n, n)$ 来编码任意长度的用户消息，并融入了可变速率特性。因此，与传统的长度为 $n$ 的分组码不同，所导出的固定分组长度 $n$（在 $\texttt{GF}(q^{n})$ 上）的多速率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——除了现有的码率 $k/n$ 之外，还包括码率 $1/n, 2/n, \ldots, (k-1)/n$。所提出的多速率编码方案还配备了一种解码策略，其中通过使用{\it 正交投影算子}的简单过程来识别可变长度的子码编码用户消息。