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Stopping set distribution of LDPC code ensembles

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3 Author(s)
Orlitsky, A. ; Dept. of Electr., Univ. of California, La Jolla, CA, USA ; Viswanathan, K. ; Junan Zhang

Stopping sets determine the performance of low-density parity-check (LDPC) codes under iterative decoding over erasure channels. We derive several results on the asymptotic behavior of stopping sets in Tanner-graph ensembles, including the following. An expression for the normalized average stopping set distribution, yielding, in particular, a critical fraction of the block length above which codes have exponentially many stopping sets of that size. A relation between the degree distribution and the likely size of the smallest nonempty stopping set, showing that for a √1-λ'(0)ρ'(1) fraction of codes with λ'(0)ρ'(1)<1, and in particular for almost all codes with smallest variable degree >2, the smallest nonempty stopping set is linear in the block length. Bounds on the average block error probability as a function of the erasure probability ε, showing in particular that for codes with lowest variable degree 2, if ε is below a certain threshold, the asymptotic average block error probability is 1-√1-λ'(0)ρ'(1)ε.

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Information Theory, IEEE Transactions on  (Volume:51 ,  Issue: 3 )