  Quantum computers will need effective error-correcting codes. Current quantum
processors require precise control of each particle, so having fewer particles
to control might be beneficial. Although traditionally quantum computers are
considered as using qubits (2-level systems), qudits (systems with more than
2-levels) are appealing since they can have an equivalent computational space
using fewer particles, meaning fewer particles need to be controlled. In this
work we prove how to construct codes with parameters $[[2^N,2^N-1-2N,\geq
3]]_q$ for any choice of prime $q$ and natural number $N$. This is accomplished
using the technique of local-dimension-invariant (LDI) codes. Generally LDI
codes have the drawback of needing large local-dimensions to ensure the
distance is at least preserved, and so this work also reduces this requirement
by utilizing the structure of CSS codes, allowing for the aforementioned code
family to be imported for any local-dimension choice.
