Creating Boolean Functions for the Five-EPR-Pair, Single-Error-Correcting Code

A quantum single-error-correcting scheme can be derived from a one-way entanglement purification protocol in purifying one Bell state from a finite block of five Bell states. The main issue to be concerned with in the theory of such an error-correction is to create specific linear Boolean functions that can transform the sixteen error syndromes occurring in the error-correcting code onto their mappings so that one Bell state is corrected whenever the other four in the finite block are measured. The Boolean function is performed under the effect of its associated sequence of basic quantum unilateral and bilateral operations. Previously, the Boolean function is created in use of the Monte Carlo computer search method. We introduce here a systematic scenario for creating the Boolean function and its associated sequence of operations so that we can do the job in an analytical way without any trial and error effort. Consequently, all possible Boolean functions can in principle be created by using our method. Furthermore, for a deduced Boolean function, we can also in the spirit of our method derive its best associated sequence of operations which may contain the least number of total operations or the least number of the bilateral XOR operations alone. Some results obtained in this work show the capability of our method in creating the Boolean function and its sequence of operations.