New quantum codes, minimum distance bounds, and equivalence of codes

Quantum error-correcting codes (quantum codes) are applied to protect quantum information from errors caused by noise (decoherence) on the quantum channel in a way that is similar to that of classical error-correcting codes. The stabilizer construction is currently the most successful and widely used technique for constructing binary quantum codes. We explore new frontiers beyond the stabilizer construction. Our approach enables integration of a broader class of classical codes into the mathematical framework of quantum stabilizer codes. Our construction is particularly well-suited to certain families of classical codes, including duadic codes and additive twisted codes. For duadic codes, we provide various modifications of our construction and develop new computational strategies to bound the minimum distance. This enabled us to extend the tables of good duadic codes to much larger block lengths. The primary focus of this thesis is on additive twisted codes, which are highly structured but also technically much more difficult than the more common families of codes. They are widely referenced but have received relatively little development in previous studies. We discover new connections between twisted codes and linear cyclic codes and provide novel lower and upper 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. Another important contribution is the development of new criteria for code equivalence within the families of linear cyclic, constacyclic, and twisted codes. We introduce novel sufficient conditions for code equivalence and classify all equivalent codes of certain lengths. We prove a recent conjecture on a necessary condition for the formula describing affine equivalence. For twisted codes, we use algebraic methods, such as group actions, to determine many codes with the same parameters. These results have practical implications, as they are useful for pruning the search for new good codes, and they enabled us to discover many new record-breaking linear and binary quantum codes.