A cutting-plane method based on redundant rows for improving fractional distance

• Published in: IEEE journal on selected areas in communications Vol. 27; no. 6; pp. 1005 - 1012
• Main Authors: Miwa, M., Wadayama, T., Takumi, I.
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.08.2009; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algorithms; Belief propagation; Cutting; Decoding; Design methodology; Fractional distance; Golay codes; Greedy algorithms; Linear code; Linear codes; Linear programming; Linear programming decoding; Mathematical analysis; Matrices; Maximum likelihood decoding; Minimum weight; Parity check codes; Polytopes; Redundant; Turbo codes
• ISSN: 0733-8716; 1558-0008
• DOI: 10.1109/JSAC.2009.090818

Decoding performance of linear programming (LP) decoding is closely related to geometrical properties of a fundamental polytope: fractional distance, pseudo codeword, etc. In this paper, an idea of the cutting-plane method is employed to improve the fractional distance of a given binary parity-check matrix. The fractional distance is the minimum weight (with respect to lscr 1 -distance) of nonzero vertices of the fundamental polytope. The cutting polytope is defined based on redundant rows of the parity-check matrix. The redundant rows are codewords of the dual code not yet appearing as rows in the parity-check matrix. The cutting polytope plays a key role to eliminate unnecessary fractional vertices in the fundamental polytope. We propose a greedy algorithm and its efficient implementation based on the cutting-plane method. It has been confirmed that the fractional distance of some parity-check matrices are actually improved by using the algorithm.