A method constructs a family of low-density-parity-check (LDPC) codes. The method includes identifying a code rate for an LDPC code in the family, identifying a protograph for the LDPC code, and constructing a base matrix for the LDPC code. The base matrix is constructed by replacing each zero in the protograph with a ‘−1’, selecting a corresponding value for an absolute shift for each one in the protograph based on constraining a number of relative shifts per column of the LDPC code to one and increasing a size of a smallest cycle in a graph of the LDPC code, and replacing each one in the protograph with the corresponding value.