This paper presents a construction of very high-rate low-density parity-check (LDPC) codes based on the incidence matrices of unital designs. Like the projective geometry and oval designs, unital designs exist with incidence matrices which are significantly rank deficient. Thus high-rate LDPC codes with a large number of linearly dependent parity-check equations can be constructed. The LDPC codes from unitals have Tanner graphs free of 4-cycles and perform well with iterative decoding, offering new LDPC codes at rates and lengths not available with existing algebraic LDPC codes.