Multiple-rate error-correcting coding scheme

Abstract

Error-correcting codes that can effectively encode and decode messages of distinct lengths while maintaining a constant blocklength are considered. It is known conventionally that a k-dimensional block code of length n defined over $\texttt {GF}(q^{n})$ is designed to encode a k-symbol user data in to an n-length codeword, resulting in a fixed-rate coding. In contrast, considering $q=p^{\lambda }$, this paper proposes two coding procedures (for the cases of $\lambda =k$ and $\lambda =n$) each deriving a multiple-rate code from existing channel codes defined over a composite field $\texttt {GF}(q^{n})$. Formally, the proposed coding schemes employ $\lambda$ codes ${\mathcal {C}}_{1}(\lambda , 1), {\mathcal {C}}_{2}(\lambda , 2), \ldots , {\mathcal {C}}_{\lambda }(\lambda , \lambda )$ defined over $\texttt {GF}(q)$ to encode user messages of distinct lengths and incorporate variable-rate feature. Unlike traditional block codes, the derived multiple-rate codes of fixed blocklength n can be used to encode and decode user messages $\mathbf{m}$ of distinct lengths $|\mathbf{m}| = 1, 2, \ldots , k, k+1, \ldots , kn$, thereby supporting a range of information rates—inclusive of the code rates $1/n^{2}, 2/n^{2},\ldots , k/n^{2}$ and $1/n, 2/n, \ldots , k/n$ ! A simple decoding procedure to the derived multiple-rate code is also given; in that, orthogonal projectors are employed for the identification of encoded user messages of variable length.