Codes over quotient rings of Lipschitz integers have recently attracted some attention. This work investigates the performance of Lipschitz integer constellations for transmission over the AWGN channel by means of the constellation figure of merit. A construction of sets of Lipschitz integers that leads to a better constellation figure of merit compared to ordinary Lipschitz integer constellations is presented. In particular, it is demonstrated that the concept of set partitioning can be applied to quotient rings of Lipschitz integers where the number of elements is not a prime number. It is shown that it is always possible to partition such quotient rings into additive subgroups in a manner that the minimum Euclidean distance of each subgroup is strictly larger than in the original set. The resulting signal constellations have a better performance for transmission over an additive white Gaussian noise channel compared to Gaussian integer constellations and to ordinary Lipschitz integer constellations. In addition, we present multilevel code constructions for the new signal constellations.
Abstract. This paper presents an introduction to the theory of generalized concatenated codes. We consider the encoding and decoding procedures of both ordinary concatenatedcodes, and generalized concatenatedcodes where block codes are used as inner codes. We give the main parameters of these codes, and by means of examples, show that under the same conditions the latter outperforms the former with respect to minimum code distance. We also heuristically describe the generalized concatenated decoding procedure. Then we show how the generalized concatenated coding ideas can be applied to encoded memoryless modulation. Such construction is often referred to as a multilevel code in literature. We consider the multilevel codes from the standpoint of generalized concatenated codes and with the help of a simple example, show how encoding and decoding procedures can be camed out. Ordinary concatenated and generalized concatenated coding schemes are considered next, using inner convolutional codes or inner modulation with memory. With the help of examples, we analyze the distance properties of such constructions.
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