How to calculate destabilizer group of toric and other codes

Some papers use the "destabilizer group" for more efficient simulations; see for example (page 3):

https://www.scottaaronson.com/showcase2/report/ted-yoder.pdf

"In addition to stabilizers, we also make use of destabilizers, introduced by Aaronson and Gottesman [10]. While the stabilizer generators can be thought of as virtual Z operators, the destabilizers can be thought of as their conjugate, virtual X operators."

Given the stabilizers $$S_i$$ of an $$[n,k,d]$$ code I know of algorithms to put them in standard form and calculate the encoded $$\bar X$$'s and $$\bar Z$$'s (also in standard forms). This gives me $$m+k+k$$ operators. The destabilizers are an additional $$m$$ operators on top of these; how are these calculated? Are there standard forms possibly related to the other generators'? Also, the $$[2d^2,2,d]$$ toric code is usually described with $$2d^2$$ stabilizers (2 of them redundant); how is the redundancy handled?