Abstract
We propose codes over the algebraic integers of two quadratic extensions of Q, namely, Q(i) and Q(\sqrt{-3}). The codes being proposed are designed to the Mannheim distance, although some properties regarding their Hamming distances are also presented, e.g., we show that all presented codes are maximum distance separable MDS. Efficient decoding algorithms are proposed to decode the codes when up to two coordinates of a transmitted code vector are affected by errors of arbitrary Mannheim weight. The Berlekamp-Massey algorithm is used for multiple error in correction. The practical interest in such Mannheim-metric codes is for their use in coded modulation schemes based on QAM-type constellations, for which neither Hamming nor Lee metric is appropriate.
Highlights
We review the background material on the theory of algebraic number fields that is necessary for understanding much of the remainder of this paper
The Berlekarnp-Massey algorithm is used for multiple error in particular its prime ideals have the form correction
The practical interest in such Mannbeim-metric codes is. for their use in coded modulation schemes based on QAM~type constellations, for which neither Hannning nor Lee metric is appropriate
Summary
We review the background material on the theory of algebraic number fields that is necessary for understanding much of the remainder of this paper. The alphabets of the codes (proposed in Sections N-A and N-B), denoted by A, are finite subsets of rings of algebraic integers A of a quadratic extension lK = Q(Vii) of
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