Question

Let $A_{1}$ and $A_{2}$ be MOLS of order $m$ and let $B_{1}$ and $B_{2}$ be MOLS of order $n$ Prove that $A_{1} \otimes B_{1}$ and $A_{2} \otimes B_{2}$ are MOLS of order $m n$.

Name: Problem 50, Chapter 10: Introductory Combinatorics
Uploaded: 2021-08-30T04:33:58-08:00
Duration: 7 min 38 s
Channel: Preeti Kumari
Description: Let A1 and A2 be MOLS of order m and let B1 and B2 be MOLS of order n Prove that A1 ⊗B1 and A2 ⊗B2 are MOLS of order m n.

   Let $A_{1}$ and $A_{2}$ be MOLS of order $m$ and let $B_{1}$ and $B_{2}$ be MOLS of order $n$ Prove that $A_{1} \otimes B_{1}$ and $A_{2} \otimes B_{2}$ are MOLS of order $m n$.

Introductory Combinatorics

Richard A. Brualdi 5th Edition

Chapter 10, Problem 50

Instant Answer

Solved by Expert Preeti Kumari

08/30/2021

Step 1

Two Latin squares $ A $ and $ B $ of order $ m $ are said to be mutually orthogonal if, when superimposed, every ordered pair $ (a, b) $ where $ a $ is from $ A $ and $ b $ is from $ B $ appears exactly once. Show more…

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Let $A_{1}$ and $A_{2}$ be MOLS of order $m$ and let $B_{1}$ and $B_{2}$ be MOLS of order $n$ Prove that $A_{1} \otimes B_{1}$ and $A_{2} \otimes B_{2}$ are MOLS of order $m n$.

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Key Concepts

Latin Square

A Latin square of order n is an n×n array filled with n different symbols, each occurring exactly once in each row and exactly once in each column. This fundamental combinatorial object underlies many topics in design theory and algebraic structures.

Mutually Orthogonal Latin Squares (MOLS)

MOLS are a set of Latin squares of the same order in which, when any two are superimposed, each ordered pair of symbols appears exactly once. They are an important concept in combinatorial design theory because they provide a way to construct pairwise distinct arrangements with balanced properties.

Tensor (Kronecker) Product

The tensor or Kronecker product is an operation that combines two matrices to produce a larger matrix. In the context of Latin squares, it is used to construct a larger Latin square from smaller ones, preserving properties such as orthogonality. This concept is central to the construction of new combinatorial designs from existing ones.

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