Structures and Constructions of Subsystem Codes over Finite Fields

Quantum information processing is a rapidlymounting field that promises to accelerate the speed up ofcomputations. The field utilizes the novel fundamental rules ofquantum mechanics such as accelerations. Quantum states carry-ing quantum information are tempted to noise and decoherence,that’s why the field of quantum error control comes. In thispaper, we investigate various aspects of the general theory ofquantum error control subsystem codes. Particularly, we firstestablish two generic methods to derive subsystem codes fromclassical codes that are defined over finite fields Fq and Fq2 .Second, we derive cyclic subsystem codes and using our twomethods, we derive all classes of subsystem codes. Consequently,we construct two famous families of cyclic subsystem BCH andRS codes. Cyclic subsystem RS codes are turned out to beOptimal and MDS codes saturating the singleton bound withequality. Third, we demonstrate several methods of subsystemcode constructions by extending, shortening and combining givensubsystem codes. Finally, we present tables of upper and lowerbounds on subsystem codes parameters .