A.A. Batenkov, K.A. Batenkov

Different signal ensembles (Euclidean codes) ensure various noise-immunity under identical method receiving. It is explained location feature of domain walls surrounding each of points. Optimization problem reduces to search of such signal points location that point areas have over-all, near sizes of one to another and approach to circle forms in case of continuous-amplitude channel model with additive white Gaussian noise. For all that task solution comes to problem of identical spheres closest packing at given volume well-known in multidimensional geometry. Such location guarantees the same error probability for any signal (signal areas are identical) and minimal average signal energy (areas are the most compact). It is necessary to note that optimal ensemble search problem may reduce to closest packing task by signals energy li-mitations reform even under subchannels gain factors maldistribution. There are several signal ensembles methods forming for channel model involved. They are conditionally divided on two groups: 1. methods on known closest packing basis; 2. methods on numerical optimization computing basis. Coder optimization problem (signal ensemble points forming) is formulated for optimal decoder (decision rule) in general formulation in paper. It is necessary to note that task solution depend not only on messages formed source and decoder decision rule but on channel model and al-so coder function associating with messages order the signal ensemble points vector. Loss function application in quadratic form leads signal set forming problem to continuous air. Nevertheless multiple integration along observa-tion space renders impossible optimization at large dimension since objective functional primitive in general form does not exist. Given problem can simplify by change from multiple integration to sum according to all signal constellation points in case of Gaussian noise and equiprobable message source. The claim denoting solutions identity of tasks about sphere closest packing in multidimensional space and received functional extremum seeking with average signal power constraint is formulate and proved in paper for the purpose of such approach substantiation. Signal set forming algorithm for discrete memoryless communication channel with additive white Gaussian noise is developed on the base of formulated task solution by methods of gradient search and penalty functions. Algorithm performance check is executed by several multidimensional signal set forming. Received constellations effectiveness turns out con-gruent with optimal lattice described another authors. Besides constellations received by means of algorithm allow to increase communications efficiency factors due to signal dimension expansion.