Some papers use the "destabilizer group" for more efficient simulations; see for example (page 3):
"In addition to stabilizers, we also make use of destabilizers, introduced by Aaronson and Gottesman . 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?