标题： 低密度奇偶校验码的最小伪权重和最小伪码字 显示英文标题

标题： Minimum Pseudo-Weight and Minimum Pseudo-Codewords of LDPC Codes

摘要： 在本文中，我们研究了低密度奇偶校验（LDPC）码在线性规划（LP）解码下的最小伪权重和最小伪码字。 首先，我们证明了对于围长大于4的LDPC码的伪码字的伪权重的Kelly、Sridhara、Xu和Rosenthal下界，当且仅当该伪码字是码字的真实倍数时才是紧的。 然后，我们证明了Kashyap和Vardy对LDPC码的停止距离的下界也是具有围长4的该LDPC码的伪码字的伪权重的下界，并且这个下界当且仅当该伪码字是码字的真实倍数时才是紧的。 利用这些结果，我们进一步证明了对于某些LDPC码，除了最小码字的真实倍数外，没有其他最小伪码字。 这意味着对于这些LDPC码，LP解码在信号噪声比趋于无穷大时，其解码错误概率与最大似然解码的错误概率之比趋近于1，即LP解码渐近最优。 最后，列出了一些LDPC码来说明这些结果。 显示英文摘要

摘要： In this correspondence, we study the minimum pseudo-weight and minimum pseudo-codewords of low-density parity-check (LDPC) codes under linear programming (LP) decoding. First, we show that the lower bound of Kelly, Sridhara, Xu and Rosenthal on the pseudo-weight of a pseudo-codeword of an LDPC code with girth greater than 4 is tight if and only if this pseudo-codeword is a real multiple of a codeword. Then, we show that the lower bound of Kashyap and Vardy on the stopping distance of an LDPC code is also a lower bound on the pseudo-weight of a pseudo-codeword of this LDPC code with girth 4, and this lower bound is tight if and only if this pseudo-codeword is a real multiple of a codeword. Using these results we further show that for some LDPC codes, there are no other minimum pseudo-codewords except the real multiples of minimum 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 to 1 as the signal-to-noise ratio leads to infinity. Finally, some LDPC codes are listed to illustrate these results.