#### An error event sensitive trade-off between rate and coding gain in MIMO MAC

##### Toni Ernvall, Jyrki Lahtonen, Hsiao-feng, Lu, Roope Vehkalahti

This work considers space-time block coding for the Rayleigh fading multiple-input multiple-output (MIMO) multiple access channel (MAC). If we suppose that the receiver is performing joint maximum-likelihood (ML) decoding, optimizing a MIMO MAC code against a fixed error event leads to a situation where the joint codewords of the users in error can be seen as a single user MIMO code. In such a case pair-wise error probability (PEP) based determinant criterion of Tarokh et al. can be used to upper bound the error probability. It was already proven by Lahtonen et al. that irrespective of the used codes the determinants of the differences of codewords of the overall codematrices will decay as a function of the rates of the users. This work will study this decay phenomenon further and derive upper bounds for the decay of determinants corresponding any error event. Lower bounds for the optimal decay are studied by constructions based on algebraic number theory and Diophantine approximation. For some error profiles the constructed codes will be proven to be optimal. While the perspective of the paper is that of PEP, the final part of the paper proves how the achieved decay results can be turned into statements about the diversity-multiplexing gain trade-off (DMT).

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