Constructions of balanced semi-latin rectangles with block size three

Semi-Latin rectangles (SLR) are row-column designs where each intersection contains the same number of experimental units, denoted as k >1. Moreover, each treatment appears an equal number of times in each row (n_r, say) and in each column (n_c, say) (n_r > 1 and n_c > 1 may or may not be equal). Balanced Semi-Latin rectangles (BSLR) form a subset of SLR, extending the concepts of Latin squares and Semi-Latin squares (SLS). These designs are applicable in various agricultural and industrial experiments, particularly in scenarios where one effect is considered a column effect and the other a row effect, with each intersection accommodating exactly 3 units. This article proposed two methods for constructing BSLR designs with a block size of 3. Additionally, an R package has been developed for generating these designs.