New 2--critical sets in the abelian 2--group

In this paper we determine a class of critical sets in the abelian {2--group} that may be obtained from a greedy algorithm. These new critical sets are all 2--critical (each entry intersects an intercalate, a trade of size 4) and completes in a top down manner.Comment: 25 page

Switching codes and designs

AbstractVarious local transformations of combinatorial structures (codes, designs, and related structures) that leave the basic parameters unaltered are here unified under the principle of switching. The purpose of the study is threefold: presentation of the switching principle, unification of earlier results (including a new result for covering codes), and applying switching exhaustively to some common structures with small parameters

Critical sets of full Latin squares

This thesis explores the properties of critical sets of the full n-Latin square and related combinatorial structures including full designs, (m,n,2)-balanced Latin rectangles and n-Latin cubes. In Chapter 3 we study known results on designs and the analogies between critical sets of the full n-Latin square and minimal defining sets of the full designs. Next in Chapter 4 we fully classify the critical sets of the full (m,n,2)-balanced Latin square, by describing the precise structures of these critical sets from the smallest to the largest. Properties of different types of critical sets of the full n-Latin square are investigated in Chapter 5. We fully classify the structure of any saturated critical set of the full n-Latin square. We show in Theorem 5.8 that such a critical set has size exactly equal to n³ - 2n² - n. In Section 5.2 we give a construction which provides an upper bound for the size of the smallest critical set of the full n-Latin square. Similarly in Section 5.4, another construction gives a lower bound for the size of the largest non-saturated critical set. We conjecture that these bounds are best possible. Using the results from Chapter 5, we obtain spectrum results on critical sets of the full n-Latin square in Chapter 6. In particular, we show that a critical set of each size between (n - 1)³ + 1 and n(n - 1)² + n - 2 exists. In Chapter 7, we turn our focus to the completability of partial k-Latin squares. The relationship between partial k-Latin squares and semi-k-Latin squares is used to show that any partial k-Latin square of order n with at most (n - 1) non-empty cells is completable. As Latin cubes generalize Latin squares, we attempt to generalize some of the results we have established on k-Latin squares so that they apply to k-Latin cubes. These results are presented in Chapter 8