Two classic “phase transitions” in discrete mathematics are the emergence of a giant component in a random graph as the density of edges increases, and the transition of a rand...
: ForasubsetS ofpositiveintegerslet (n, S)bethesetofpartitionsofnintosummands that are elements of S. For every (n, S), let Mn() be the number of parts, with multiplicity, that ...
In [6], Gallager introduces a family of codes based on sparse bipartite graphs, which he calls low-density parity-check codes. He suggests a natural decoding algorithm for these c...
Michael Luby, Michael Mitzenmacher, Mohammad Amin ...
Ensembles of regular low-density parity-check codes over any finite Abelian group G are studied. The nonzero entries of the parity matrix are randomly chosen, independently and uni...
Consider independent and identically distributed exponential random variables (r.vs) 1, 2, . . . , and positive scalars 1, 2, . . . , . In this letter, we present the probability...