We study communication over atmospheric turbulence channels based on LDPC-coded, multidimensional OAM signal constellations. Multidimensional signal constellation is obtained as the Cartesian product of one-dimensional signal constellation X={( i-1)d, i=1,2,...,M} ( where d is the Euclidean distance between neighboring signal constellation points and M is the number of amplitude levels) as X-N={(x(1), x(2),..., x(N))vertical bar x(i) is from X, for every i}. This scheme represents an energy efficient alternative, since log(2)(M (N)) bits/symbol can be transmitted. We describe two possible implementations of N-dimensional OAM modulator and demodulator: (1) volume holograms based, and (2) multimode fibers (MMFs) based. We evaluate the performance of this scheme by determining conditional symbol probability density functions (PDFs) from numerical propagation data. Two cases of practical interest are studied: (i) when conditional PDFs are known on the receiver side, and (ii) when conditional PDFs are not known and Gaussian approximation is used instead. We show that in case (ii) an early error floor occurs because of inaccurate PDF assumption, which is caused by OAM crosstalk introduced by the atmospheric turbulence. We also show that the OAM modulation is more sensitive to atmospheric turbulence as the number of dimensions increases. Finally, we evaluate the BER performance for different amplitude levels and different number of OAM dimensions.