We provide a new construction of quantum codes that enables integration of a broader class of classical codes into the mathematical framework of quantum stabilizer codes. Next, we present new connections between twisted codes and linear cyclic codes and provide novel bounds for the minimum distance of twisted codes. We show that classical tools such as the Hartmann–Tzeng minimum distance bound are applicable to twisted codes. This enabled us to discover five new infinite families and many other examples of record-breaking, and sometimes optimal, binary quantum codes. We also discuss the role of the γ value on the parameters of twisted codes and present new results regarding the construction of twisted codes with different γ values but identical parameters. Finally, we list many new record-breaking binary quantum codes that we obtained from additive twisted, linear cyclic, and constacyclic codes.