Superposition Construction of $Q$ -Ary LDPC Codes by Jointly Optimizing Girth and Number of Shortest Cycles

In this letter, we consider the construction of q-ary low-density parity-check (LDPC) codes by jointly optimizing the girth and number of shortest cycles in the frame of superposition. We begin with constructing binary quasi-cyclic LDPC codes by superposition. By replacing the nonzero entries of the parity-check matrices of the resulting binary codes with the nonzero elements of GF(q), a class of q-ary LDPC codes is obtained. Furthermore, we analyze the cycle structure of the constructed codes for a given degree distribution and code length, and present a search algorithm for finding q-ary LDPC codes with few cycles of length g, where g is the optimized girth value. Simulation results show that the proposed codes perform well under the iterative decoding algorithms.