Latin square designs are widely used in comparative experiments where two crossed blocking factors are present and each blocking factor has v levels, where v is also equal to the number of Latin letters in the Latin square or the number of treatments in the design. In this arrangement each Latin letter appears in each row and each column precisely once. However, it may not always be possible to accommodate all the v Latin letters or treatments precisely once in each row and / or each column, leading thereby to a situation where each row and / or each column may have less than v Latin letters or treatments appearing in them. In other words, there would be empty nodes in the Latin square arrangement and the Latin square design, meaning thereby that the row-column design is a non-orthogonal design with treatments vs rows and / or treatments vs columns classifications as non-orthogonal. Balanced incomplete Latin square designs have been introduced for such situations. A balanced incomplete Latin square design with parameters v and r is an incomplete Latin square of order v such that each row and each column has r < v non-empty cells and v − r empty cells and each of the v symbols appears exactly r times in the whole square. Henceforth, we shall denote a balanced incomplete Latin square with v symbols and r replications of each symbol as BILS (v, r). Here the term ‘balanced’ implies that each row and column has same number of non-empty cells and each symbol has same number of replications in the whole square. This balance is neither related to pair-wise balance nor variance balance. Here, construction of BILS (v, r) is done by removing the v − r disjoint transversals from a Latin square of order v via a pair of orthogonal Latin squares (Ai et al., 2013). Here, a transversal in a Latin Square of order v is a set of v cells such that only one cell is allowed in each row and in each column, and furthermore, each symbol can appear in each cell once.

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