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In a digital communication system, information is sent from one place toanother over a noisy communication channel using binary symbols (bits).Original information is encoded by adding redundant bits, which are then usedby low--density parity--check (LDPC) codes to detect and correct errors thatmay have been introduced during transmission. Error correction capability of anLDPC code is severely degraded due to harmful structures such as small cyclesin its bipartite graph representation known as Tanner graph (TG). We introducean integer programming formulation to generate a TG for a given smallest cyclelength. We propose a branch-and-cut algorithm for its solution and investigatestructural properties of the problem to derive valid inequalities and variablefixing rules. We introduce a heuristic to obtain feasible solutions of theproblem. Our computational experiments show that our algorithm can generateLDPC codes without small cycles in acceptable amount of time for practicallyrelevant code lengths.

A Branch-and-Cut Algorithm to Design LDPC Codes without Small Cycles in Communication Systems