This paper proposes the periodical successive over-relaxation (PSOR)-Jacobi algorithm for minimum mean squared error (MMSE) detection of multiple-input multiple-output (MIMO) signals. The proposed algorithm has the advantages of two conventional methods. One is the Jacobi method, which is an iterative method for solving linear equations and is suitable for parallel implementation. The Jacobi method is thus a promising candidate for high-speed simultaneous linear equation solvers for the MMSE detector. The other is the Chebyshev PSOR method, which has recently been shown to accelerate the convergence speed of linear fixed-point iterations. We compare the convergence performance of the PSOR-Jacobi algorithm with that of conventional algorithms via computer simulation. The results show that the PSOR-Jacobi algorithm achieves faster convergence without increasing computational complexity, and higher detection performance for a fixed number of iterations. This paper also proposes an efficient computation method of inverse matrices using the PSOR-Jacobi algorithm. The results of computer simulation show that the PSOR-Jacobi algorithm also accelerates the computation of inverse matrix.

Recently, Haley and Grant introduced the concept of reversible codes – a class of binary linear codes that can reuse the decoder architecture as the encoder and encodable by the iterative message-passing algorithm based on the Jacobi method over F2. They also developed a procedure to construct parity check matrices of a class of reversible codes named type-I reversible codes by utilizing properties specific to circulant matrices. In this paper, we refine a mathematical framework for reversible codes based on circulant matrices through a ring theoretic approach. This approach enables us to clarify the necessary and sufficient condition on which type-I reversible codes exist. Moreover, a systematic procedure to construct all circulant matrices that constitute parity check matrices of type-I reversible codes is also presented.