Completly Randomized Trials versus Incomplete Cubic Lattice

I have a 2x2x2x2 design, which I wish to balance. Previously, the binary factors appeared throughout the experiment with a 50% probability, which each have to result in a uniform distribution. I am only interested in the effect of two factors. (One of the binary factors switches the response pattern, while the other binary factor is used to compute a difference). In addition, two stochastic processes determine each treatment.

Because I wish to avoid a serial order effect in this psychological experiment, it is desirable that trials appear in random order. However, I have been reading about cubic lattices and so, I have asked myself what the advantage would be with an incomplete cubic lattice blocked design, which we randomize across subjects. A complete design would be feasible, if my computations are correct $2^{8}=256$ and we aim for approximately 1000 trials, say 1024.

I am only interested in the 2×2 factors (i.e. $2^{2}=4$). Is it possible to balance the design and randomize block order across participants? If so, what would be the advantages/disadvantages?