Multiple-Rate Channel Codes in GF(p^n^2)
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A code C(n, k, d) defined over 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 GF(q^n). Formally, by viewing a symbol of GF(q^n) as an n-tuple over the base field GF(q), the proposed coding scheme employs children codes C_1(n, 1), C_2(n, 2), ..., C_n(n, n) defined over 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 GF(q^n)) can be used to encode and decode user messages m (over GF(q)) of arbitrary lengths | m| = k, k+1, ..., kn, thereby supporting a range of information rates - inclusive of the code rates 1/n, 2/n, ..., (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 orthogonal projectors.
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