Ring Theoretic Approach to Reversible Codes Based on Circulant Matrices

Tomoharu SHIBUYA

IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E94-A    No.11    pp.2121-2126
Publication Date: 2011/11/01
Online ISSN: 1745-1337
DOI: 10.1587/transfun.E94.A.2121
Print ISSN: 0916-8508
Type of Manuscript: Special Section PAPER (Special Section on Information Theory and Its Applications)
Category: Coding Theory
LDPC codes,  reversible codes,  encoding of linear codes,  Jacobi method,  circulant matrices,  message-passing algorithm,  

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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.