New Coding Techniques for Codes over Gaussian Integers

This work presents block codes over Gaussian integers. We introduce Gaussian integer rings which extend the number of possible signal constellations over Gaussian integer fields. Many well-known code constructions can be used for codes over Gaussian integer rings, e.g., the Plotkin construction or product codes. These codes enable low complexity decoding in the complex domain. Furthermore, we demonstrate that the concept of set partitioning can be applied to Gaussian integers. This enables multilevel code constructions. In addition to the code constructions, we present a low complexity soft-input decoding algorithm for one Mannheim error correcting codes. The presented decoding method is based on list decoding, where the list of candidate codewords is obtained by decomposing the syndrome into two sub-syndromes. Considering all decompositions of the syndrome we construct lists of all possible errors of Mannheim weight two. In the last decoding step the squared Euclidean distance is used to select the best codeword from the list. Simulation results for the additive white Gaussian noise channel demonstrate that the proposed decoding method achieves a significant coding gain compared with hard-input decoding.