Error-correcting codes are fundamental in modern digital communication with applications in data storage and data transmission. Interest in a class of error-correcting codes called low-density parity-check (LDPC) codes has been growing since their recent rediscovery because of their low decoding complexity and their high-performance. However, practical applications have been limited due to the difficulty of finding good LDPC codes for practical parameters. This paper proposes an exhaustive and a randomized algorithm for constructing a family of LDPC codes with practical parameters whose matrix representations meet the following requirements: for each row in the LDPC code matrix there exists exactly one common nonzero element, each row has a minimum weight of one and must be odd, and each column has a weight of at least two. These conditions improve performance of the resulting codes and simplify conversion into codes for quantum systems. Both algorithms utilize a maximal clique algorithm to construct LDPC matrices from graphs whose vertices are possible rows within said matrices and are adjacent the first condition is true. While these algorithms were found to be suitable for small parameters, future work which optimizes the resulting codes for their expected applications could also dramatically increase performance of the algorithms themselves.