Two matrices are said to be non-overlapping if there is no way to put one of them on the other one such that the top-left corner of one matrix coincides with the bottom-right corner of another matrix, or the top-right corner of one matrix coincides with the bottom-left corner of another matrix. A two-dimensional non-overlapping code is a set of \(m\times n\) matrices (called codewords) over a q-ary alphabet if every pair of matrices within the set (whether identical or not) are non-overlapping. We provide a new construction for two-dimensional q-ary non-overlapping codes and compare sizes of our codes and those in the literature. Especially, the set of binary non-overlapping matrices obtained from our construction can be listed in a Gray code sense. We also initialize the study on two-dimensional non-expandable non-overlapping codes over a q-ary alphabet for any \(q>2\) by generalizing the two-dimensional binary non-expandable non-overlapping codes in the literature.
