Checkered patterns are characterized by their square structure and the use of only two distinct colors. These colors are typically represented by two types of numerical sets: { 1,0} and { 1,-1}. Matrices based on { 1,0} may seem identical to those based on { 1,-1} when forming checkered patterns because the only difference is that the numbers 0 are changed to -1. However, these two kinds of matrices are completely different in a mathematical sense because a matrix using { 1,0} has a rank of 2 and a matrix using { 1,-1} has a rank of 1. Knowing this difference in advance allows us to reduce the computational effort required for matrix operations such as information embedding and retrieving.
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