On the optimality of extended maximal length linear feedback shift register sequences

•We provide sufficient conditions for functional MRI designs to be optimal.•We present and prove the optimality of an extended mm-sequence.•An extended mm-sequence is universally optimal for studies with one stimulus type.•An extended mm-sequence is DD-optimal for cases with two or more stimulus types.

Maximal length linear feedback shift register sequences (or mm-sequences) are widely used as experimental designs for functional magnetic resonance imaging (fMRI) experiments. However, the statistical optimality of these designs has not been proven. Here, we present and study an extended version of mm-sequences and show that these designs are optimal.