Reed-Solomon (RS) and Bose-Chaudhuri-Hocquenghem (BCH) error correcting codes are widely used in digital technology. An important problem in the implementation of RS and BCH decoding is the fast finding of the error positions (the roots of error locator polynomials). Several fast root-finding algorithms for polynomials over finite fields have been proposed. In this paper we give a generalization of the Goertzel algorithm. Our algorithm is suitable for the parallel hardware implementation and the time of multiplications used is restricted by a constant.
In quantum circuits, importing of additional qubits can reduce the operation time and prevent decoherence induced by the environment. However, excessive qubits may make the quantum system vulnerable. This paper describes how to relax existing qubits without additional qubits to significantly reduce the operation time of the quantum Fourier circuit compared to a circuit without optimization. The results indicate that this scheme makes full use of the qubits relaxation. The concepts can be applied to improve similar quantum circuits and guide the physical implementations of quantum algorithms or devices.
