Generalization of Cycle Decompositions of Even Dimensional Hypercubes on $d$-Dimensional Toruses

We consider cycle decompositions of even, $2an$-dimensional hypercubes $Q_{2an},$ where $a \geq 3$ is odd and $n \geq 1.$ Prior work done by Axenovich, Offner, and Tompkins focused on obtaining the existence of cycle decompositions for even-dimensional hypercubes using long cycles of a given form, leaving out cycles of shorter lengths and, in fact, cycles of even longer lengths than those obtained there, such as $C_{7 \cdot 2^{11}}$ in the case of $Q_{14}.$ In this paper, we provide two novel methods for explicitly constructing cycle decompositions of virtually all possible cycle lengths, using cycles of a given form, on Cartesian products of cycles up to those known by the work of Axenovich, Offner, and Tompkins. In particular, we show that we can explicitly obtain cycle decompositions of even dimensional hypercubes $Q_{2an}$ for all lengths mentioned above while on the same Cartesian product of cycles. With this, the current understanding of cycle decompositions of even dimensional hypercubes is furthered constructively and is featured with some interesting consequences for when $a$ is a positive, even integer. Additionally, progress is made towards obtaining cycle decompositions using the longest admissible cycle lengths with the incorporation of a more explicit starting point from which such decompositions of $Q_{2an}$ can be studied further.