Various embodiments of the present invention provide systems and methods for generating a code format. One method discussed includes: receiving a first matrix having a row width and a column height that is greater than one; incorporating a circulant into a first column of the first matrix; testing the first column for trapping sets, wherein at least one trapping set is identified; selecting a value to mitigate the identified trapping set; and augmenting the first matrix with a second matrix to yield a composite matrix. The second matrix has the selected value in the first column, and wherein the identified trapping set is mitigated.