面向哆维信号且具有代数译码算法的纠错码

The demand for higher data rate and power efficiencies in digital communications.makes multidimensional modulation schemes increasingly attractive. Motivation for.the use of multidimensional signals dates back to the work of Shannon. In his.celebrated analysis of the limit performance achievable in digital communication over a given channel, he recognized that the performance of a signal constellation used to transmit digital information over the additive white Gaussian noise channel can be.improved by increasing the dimensionality of the signal set used for transmission. In particular, as the dimension number grows to infinity, the performance tends to an upper limit that defines the apacity of the channel. In the last two decades, many results about multidimensional modulation and multidimensional trellis codes have been obtained. However, little is known about linear block codes over finite fields for coding multidimensional signals. It is of important to construct error-correcting codes with algebraic decoding algorithms in computer and communication areas since the.performances of these codes can be analyed with mathematical tools. This project.intends to do basic research around important problems of communications. The.innovation is to develop study by combining the basic theory of algebraic number theory , the principals and methods of error-correcting codes and computer techniques. The research results we have obtained include the following: It is showed that one subgroup of the multiplicative group of units in the.algebraic integer ring of each quadratic number field with unique factorization.property $Q(\sqrt {m})$, modulo the ideal $ (2^{n})$, can be used to obtain a.QAM signal space of $2^{2n-2}$ points with good geometrical properties, where.$n\geq 3$, $m \not \equiv 1 ({\rm mod}~8)$ and $m$ is a square-free rational integer. These QAM signals can be coded such that a differentially coherent method can be applied to demodulate the QAM signals. The multiplicative subgroups can also be used to construct block codes over Gaussian integers which are able to correct some error patterns. The result has been published in “IMA Journalof Mathematical Control and Information”.The linear codes are constructed by using algebraic integer rings of yclotomic fields such that these codes are for multidimensional signals with algebraic decoding algorithms. Some results on residue codes and the norm quadratic-residue codes are also obtained.

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