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Design of low-density parity-check (LDPC) codes suitable for all channels which exhibit a given capacity C is investigated. Such codes are referred to as universal LDPC codes. First, based on numerous observations, a conjecture is put forth that a code working on N equal-capacity channels, also works on any convex combination of these N channels. As a supporting evidence, we prove that a code satisfying the stability condition on N channels, also satisfies the stability condition on the convex hull of these N channels. Then, a channel decomposition method is suggested which spans any given channel with capacity C in terms of a number of identical-capacity basis channels. We expect codes that work on the basis channels to be suitable for any convex combination of the bases, i.e., all channels with capacity C. Such codes are found over a wide range of rates. An upper bound on the achievable rate of universal LDPC codes is suggested. Through examples, it is shown that our codes achieve rates extremely close to this upper bound. In comparison with existing LDPC codes designed for a given channel, significant performance gain is reported when codes are used over various channels of equal capacity.