Odd-characteristic analogue of the F_{2^h}-to-binary bijection and geometry
Establish whether an analogue of the bijection between stabiliser codes over F_{2^h} and binary stabiliser codes exists for stabiliser codes over finite fields of odd characteristic, and characterize the corresponding geometric objects and equivalence-preserving operations. Address the challenges that arise from the potential nonexistence of trace-orthogonal bases in odd characteristic and from the currently unknown geometric correspondence, potentially by formulating the interpretation in terms of affine rather than projective points.
References
We solved [1, Research Problem 4] in the case when q is even. Can we do something similar for codes over finite fields of odd characteristic? There are two drawbacks. First, there does not always exist a trace-orthogonal basis in odd characteristic. Second, it has not been established what these codes correspond to geometrically. Multiplying a column with -1 does not affect the projective geometry, but it does not come from a conjugation with a Clifford operator. This suggests that, instead of looking at the projective points determined by the columns, we should rather look at them as affine points.