Ring Theoretic Approach to Reversible Codes Based on Circulant Matrices

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 F-2. 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.