Minimum Pseudoweight and Minimum Pseudocodewords of LDPC Codes
Shu-Tao Xia
Fang-Wei Fu
Tsinghua Univ., Shenzhen;
This paper appears in: Information Theory, IEEE Transactions on
Publication Date: Jan. 2008
Volume: 54,
Issue: 1
On page(s): 480-485
ISSN: 0018-9448
INSPEC Accession Number: 9742870
Digital Object Identifier: 10.1109/TIT.2007.911177
Current Version Published: 2008-01-04
Abstract
In this correspondence, we study the minimum pseudoweight and minimum pseudocodewords of low-density parity-check (LDPC) codes under linear programming (LP) decoding. First, we show that the lower bound of Kelley, Sridhara, Xu, and Rosenthal on the pseudoweight of a nonzero pseudocodeword of an LDPC code whose Tanner graph has girth greater than is tight if and only if this pseudocodeword is a real multiple of a codeword. Then, the lower bound of Kashyap and Vardy on the stopping distance of an LDPC code is proved to be also a lower bound on the pseudoweight of a nonzero pseudocodeword of an LDPC code whose Tanner graph has girth , and this lower bound is tight if and only if this pseudocodeword is a real multiple of a codeword. Using these results we further obtain that for some LDPC codes, there are no other minimum pseudocodewords except the real multiples of minimum weight codewords. This means that the LP decoding for these LDPC codes is asymptotically optimal in the sense that the ratio of the probabilities of decoding errors of LP decoding and maximum-likelihood decoding approaches as the signal-to-noise ratio (SNR) tends to infinity. Finally, some LDPC codes are listed to illustrate these results.
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