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We develop new blind and semi-blind data detectors and channel estimators for orthogonal frequency-division multiplexing (OFDM) systems. Our data detectors require minimizing a complex, integer quadratic form in the data vector. The semi-blind detector uses both channel correlation and noise variance. The quadratic for the blind detector suffers from rank deficiency; for this, we give a low-complexity solution. Avoiding a computationally prohibitive exhaustive search, we solve our data detectors using sphere decoding (SD) and V-BLAST and provide simple adaptations of the SD algorithm. We consider how the blind detector performs under mismatch, generalize the basic data detectors to nonunitary constellations, and extend them to systems with pilots and virtual carriers. Simulations show that our data detectors perform well.