When q-Steiner systems exist for given parameters, the dimension of the space they span via characteristic vectors is [n k]_q - [n t]_q + 1.
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On the dimension of the space generated by characteristic vectors of $q$-Steiner systems
When q-Steiner systems exist for given parameters, the dimension of the space they span via characteristic vectors is [n k]_q - [n t]_q + 1.