Locating arrays and disjoint partitions Daniel Horsley (Monash - PowerPoint PPT Presentation

Locating arrays Not a ( 1 , 1 ) -locating array with N = 6, k = 9 and v = 3: 1 3 3 3 2 2 1 3 2 2 1 3 3 3 2 2 1 1 2 2 1 3 3 3 1 2 3 3 2 2 1 3 3 3 1 3 3 3 2 2 1 3 2 3 1 3 3 3 2 2 1 3 2 2

Locating arrays Not a ( 1 , 1 ) -locating array with N = 6, k = 9 and v = 3: 1 3 3 3 2 2 1 3 2 2 1 3 3 3 2 2 1 1 2 2 1 3 3 3 1 2 3 3 2 2 1 3 3 3 1 3 3 3 2 2 1 3 2 3 1 3 3 3 2 2 1 3 2 2 A ( 1 , 1 ) -locating array with N = 6, k = 9 and v = 3: 1 3 3 3 2 2 1 3 2 2 1 3 3 3 2 2 1 3 2 2 1 3 3 3 1 2 1 3 2 2 1 3 3 3 1 2 3 3 2 2 1 3 2 3 1 3 3 3 2 2 1 3 2 3

Locating arrays Not a ( 1 , 1 ) -locating array with N = 6, k = 9 and v = 3: 1 3 3 3 2 2 1 3 2 2 1 3 3 3 2 2 1 1 2 2 1 3 3 3 1 2 3 3 2 2 1 3 3 3 1 3 3 3 2 2 1 3 2 3 1 3 3 3 2 2 1 3 2 2 A ( 1 , 1 ) -locating array with N = 6, k = 9 and v = 3: 1 3 3 3 2 2 1 3 2 2 1 3 3 3 2 2 1 3 2 2 1 3 3 3 1 2 1 3 2 2 1 3 3 3 1 2 3 3 2 2 1 3 2 3 1 3 3 3 2 2 1 3 2 3 Given k and v , we want to minimise N .

Our problem: Given N and v , find the maximum number of columns in a ( 1 , 1 ) -locating array with N rows and v symbols.

Our problem: Given N and v , find the maximum number of columns in a ( 1 , 1 ) -locating array with N rows and v symbols. Similar problems have been considered in the combinatorial group testing literature, but not this one exactly.

Disjoint set partitions

Disjoint set partitions

Disjoint set partitions 1 3 3 3 2 2 1 3 2 2 1 3 3 3 2 2 1 3 2 2 1 3 3 3 1 2 1 3 2 2 1 3 3 3 1 2 3 3 2 2 1 3 2 3 1 3 3 3 2 2 1 3 2 3

Disjoint set partitions 1 3 3 3 2 2 1 3 2 2 1 3 3 3 2 2 1 3 2 2 1 3 3 3 1 2 1 3 2 2 1 3 3 3 1 2 3 3 2 2 1 3 2 3 1 3 3 3 2 2 1 3 2 3 Equivalently: { 1 } { 2 , 3 } { 4 , 5 , 6 } { 2 } { 3 , 4 } { 1 , 5 , 6 } { 3 } { 4 , 5 } { 1 , 2 , 6 } { 4 } { 5 , 6 } { 1 , 2 , 3 } { 5 } { 1 , 6 } { 2 , 3 , 4 } { 6 } { 1 , 2 } { 3 , 4 , 5 } { 1 , 3 } { 2 , 5 } { 4 , 6 } { 2 , 4 } { 3 , 6 } { 1 , 5 } { 3 , 5 } { 1 , 4 } { 2 , 6 }

Disjoint set partitions 1 3 3 3 2 2 1 3 2 2 1 3 3 3 2 2 1 3 2 2 1 3 3 3 1 2 1 3 2 2 1 3 3 3 1 2 3 3 2 2 1 3 2 3 1 3 3 3 2 2 1 3 2 3 Equivalently: { 1 } { 2 , 3 } { 4 , 5 , 6 } { 2 } { 3 , 4 } { 1 , 5 , 6 } { 3 } { 4 , 5 } { 1 , 2 , 6 } { 4 } { 5 , 6 } { 1 , 2 , 3 } { 5 } { 1 , 6 } { 2 , 3 , 4 } { 6 } { 1 , 2 } { 3 , 4 , 5 } { 1 , 3 } { 2 , 5 } { 4 , 6 } { 2 , 4 } { 3 , 6 } { 1 , 5 } { 3 , 5 } { 1 , 4 } { 2 , 6 }

Disjoint set partitions

Disjoint set partitions Fact A ( 1 , 1 ) -locating array with N rows, k columns and v symbols is equivalent to k partitions of { 1 , . . . , N } , each with v nonempty classes, such that no two of the kv classes are equal.

Disjoint set partitions Fact A ( 1 , 1 ) -locating array with N rows, k columns and v symbols is equivalent to k partitions of { 1 , . . . , N } , each with v nonempty classes, such that no two of the kv classes are equal. I’ll say disjoint v-partitions of { 1 , . . . , N } from now on.

Disjoint set partitions Fact A ( 1 , 1 ) -locating array with N rows, k columns and v symbols is equivalent to k partitions of { 1 , . . . , N } , each with v nonempty classes, such that no two of the kv classes are equal. I’ll say disjoint v-partitions of { 1 , . . . , N } from now on. Our problem (rephrased) Given N and v , find the maximum number of disjoint v -partitions of { 1 , . . . , N } .

Disjoint set partitions Fact A ( 1 , 1 ) -locating array with N rows, k columns and v symbols is equivalent to k partitions of { 1 , . . . , N } , each with v nonempty classes, such that no two of the kv classes are equal. I’ll say disjoint v-partitions of { 1 , . . . , N } from now on. Our problem (rephrased) Given N and v , find the maximum number of disjoint v -partitions of { 1 , . . . , N } .

Disjoint set partitions Fact A ( 1 , 1 ) -locating array with N rows, k columns and v symbols is equivalent to k partitions of { 1 , . . . , N } , each with v nonempty classes, such that no two of the kv classes are equal. I’ll say disjoint v-partitions of { 1 , . . . , N } from now on. Our problem (rephrased) Given N and v , find the maximum number of disjoint v -partitions of { 1 , . . . , N } .

Disjoint set partitions Fact A ( 1 , 1 ) -locating array with N rows, k columns and v symbols is equivalent to k partitions of { 1 , . . . , N } , each with v nonempty classes, such that no two of the kv classes are equal. I’ll say disjoint v-partitions of { 1 , . . . , N } from now on. Our problem (rephrased) Given N and v , find the maximum number of disjoint v -partitions of { 1 , . . . , N } . Similar problems have been considered in the set systems literature, but not this one exactly.

Shapes

Shapes { 1 } { 2 , 3 } { 4 , 5 , 6 } { 2 } { 3 , 4 } { 1 , 5 , 6 } { 3 } { 4 , 5 } { 1 , 2 , 6 } { 4 } { 5 , 6 } { 1 , 2 , 3 } { 5 } { 1 , 6 } { 2 , 3 , 4 } { 6 } { 1 , 2 } { 3 , 4 , 5 } { 1 , 3 } { 2 , 5 } { 4 , 6 } { 2 , 4 } { 3 , 6 } { 1 , 5 } { 3 , 5 } { 1 , 4 } { 2 , 6 }

Shapes partition shape { 1 } { 2 , 3 } { 4 , 5 , 6 } [1,2,3] { 2 } { 3 , 4 } { 1 , 5 , 6 } [1,2,3] { 3 } { 4 , 5 } { 1 , 2 , 6 } [1,2,3] { 4 } { 5 , 6 } { 1 , 2 , 3 } [1,2,3] { 5 } { 1 , 6 } { 2 , 3 , 4 } [1,2,3] { 6 } { 1 , 2 } { 3 , 4 , 5 } [1,2,3] { 1 , 3 } { 2 , 5 } { 4 , 6 } [2,2,2] { 2 , 4 } { 3 , 6 } { 1 , 5 } [2,2,2] { 3 , 5 } { 1 , 4 } { 2 , 6 } [2,2,2]

Shapes partition shape { 1 } { 2 , 3 } { 4 , 5 , 6 } [1,2,3] { 2 } { 3 , 4 } { 1 , 5 , 6 } [1,2,3] { 3 } { 4 , 5 } { 1 , 2 , 6 } [1,2,3] { 4 } { 5 , 6 } { 1 , 2 , 3 } [1,2,3] { 5 } { 1 , 6 } { 2 , 3 , 4 } [1,2,3] { 6 } { 1 , 2 } { 3 , 4 , 5 } [1,2,3] { 1 , 3 } { 2 , 5 } { 4 , 6 } [2,2,2] { 2 , 4 } { 3 , 6 } { 1 , 5 } [2,2,2] { 3 , 5 } { 1 , 4 } { 2 , 6 } [2,2,2] k disjoint v -partitions of { 1 , . . . , N } give rise to k shapes , each with v parts, � N � such at most of the kv parts are equal to i for i ∈ { 1 , . . . , N } . i

Shapes partition shape { 1 } { 2 , 3 } { 4 , 5 , 6 } [1,2,3] { 2 } { 3 , 4 } { 1 , 5 , 6 } [1,2,3] { 3 } { 4 , 5 } { 1 , 2 , 6 } [1,2,3] { 4 } { 5 , 6 } { 1 , 2 , 3 } [1,2,3] { 5 } { 1 , 6 } { 2 , 3 , 4 } [1,2,3] { 6 } { 1 , 2 } { 3 , 4 , 5 } [1,2,3] { 1 , 3 } { 2 , 5 } { 4 , 6 } [2,2,2] { 2 , 4 } { 3 , 6 } { 1 , 5 } [2,2,2] { 3 , 5 } { 1 , 4 } { 2 , 6 } [2,2,2] k disjoint v -partitions of { 1 , . . . , N } give rise to k shapes , each with v parts, � N � such at most of the kv parts are equal to i for i ∈ { 1 , . . . , N } . i I’ll say admissible family of v-shapes from now on.

Shapes partition shape { 1 } { 2 , 3 } { 4 , 5 , 6 } [1,2,3] { 2 } { 3 , 4 } { 1 , 5 , 6 } [1,2,3] { 3 } { 4 , 5 } { 1 , 2 , 6 } [1,2,3] { 4 } { 5 , 6 } { 1 , 2 , 3 } [1,2,3] { 5 } { 1 , 6 } { 2 , 3 , 4 } [1,2,3] { 6 } { 1 , 2 } { 3 , 4 , 5 } [1,2,3] { 1 , 3 } { 2 , 5 } { 4 , 6 } [2,2,2] { 2 , 4 } { 3 , 6 } { 1 , 5 } [2,2,2] { 3 , 5 } { 1 , 4 } { 2 , 6 } [2,2,2] k disjoint v -partitions of { 1 , . . . , N } give rise to k shapes , each with v parts, � N � such at most of the kv parts are equal to i for i ∈ { 1 , . . . , N } . i I’ll say admissible family of v-shapes from now on.

Shapes

Shapes Theorem A family of disjoint partitions of { 1 , . . . , N } with specified shapes exists if and only if the family of specified shapes is admissible.

Shapes Theorem A family of disjoint partitions of { 1 , . . . , N } with specified shapes exists if and only if the family of specified shapes is admissible. This is a generalisation of: Baranyai’s theorem � uv There are 1 � disjoint v -partitions of { 1 , . . . , uv } such that each class of v u each partition has size u .

Shapes Theorem A family of disjoint partitions of { 1 , . . . , N } with specified shapes exists if and only if the family of specified shapes is admissible. This is a generalisation of: Baranyai’s theorem � uv There are 1 � disjoint v -partitions of { 1 , . . . , uv } such that each class of v u each partition has size u . Our proof is adapted from an inductive proof Baranyai’s theorem, due to Brouwer and Schrijver, that uses the integer flow theorem.

Shapes Theorem A family of disjoint partitions of { 1 , . . . , N } with specified shapes exists if and only if the family of specified shapes is admissible. This is a generalisation of: Baranyai’s theorem � uv There are 1 � disjoint v -partitions of { 1 , . . . , uv } such that each class of v u each partition has size u . Our proof is adapted from an inductive proof Baranyai’s theorem, due to Brouwer and Schrijver, that uses the integer flow theorem. See also Bahmanian and Bryant.

Shapes Theorem A family of disjoint partitions of { 1 , . . . , N } with specified shapes exists if and only if the family of specified shapes is admissible. This is a generalisation of: Baranyai’s theorem � uv There are 1 � disjoint v -partitions of { 1 , . . . , uv } such that each class of v u each partition has size u . Our proof is adapted from an inductive proof Baranyai’s theorem, due to Brouwer and Schrijver, that uses the integer flow theorem. See also Bahmanian and Bryant. Our problem (re-rephrased) Given N and v , find the maximum size of an admissible family of v -shapes.

Maximal families

Maximal families ( f = ⌊ N + 1 Example: N = 38, v = 7 v ⌋ = 5, d = v ( f + 1 ) − N = 4)

Maximal families ( f = ⌊ N + 1 Example: N = 38, v = 7 v ⌋ = 5, d = v ( f + 1 ) − N = 4) � 38 � 38 ◮ There are at most � � + special shapes containing a 1 or 2. 1 2

Maximal families ( f = ⌊ N + 1 Example: N = 38, v = 7 v ⌋ = 5, d = v ( f + 1 ) − N = 4) � 38 � 38 ◮ There are at most � � + special shapes containing a 1 or 2. 1 2 ◮ Let the defect of a part x � 5 be f + 1 − x = 6 − x .

Maximal families ( f = ⌊ N + 1 Example: N = 38, v = 7 v ⌋ = 5, d = v ( f + 1 ) − N = 4) � 38 � 38 ◮ There are at most � � + special shapes containing a 1 or 2. 1 2 ◮ Let the defect of a part x � 5 be f + 1 − x = 6 − x . � 38 � 38 � 38 ◮ The total defect in the other shapes is at most 3 � � � + 2 + . 3 4 5

Maximal families ( f = ⌊ N + 1 Example: N = 38, v = 7 v ⌋ = 5, d = v ( f + 1 ) − N = 4) � 38 � 38 ◮ There are at most � � + special shapes containing a 1 or 2. 1 2 ◮ Let the defect of a part x � 5 be f + 1 − x = 6 − x . � 38 � 38 � 38 ◮ The total defect in the other shapes is at most 3 � � � + 2 + . 3 4 5 ◮ Any shape has total defect at least d = 4.

Maximal families ( f = ⌊ N + 1 Example: N = 38, v = 7 v ⌋ = 5, d = v ( f + 1 ) − N = 4) � 38 � 38 ◮ There are at most � � + special shapes containing a 1 or 2. 1 2 ◮ Let the defect of a part x � 5 be f + 1 − x = 6 − x . � 38 � 38 � 38 ◮ The total defect in the other shapes is at most 3 � � � + 2 + . 3 4 5 ◮ Any shape has total defect at least d = 4. � 38 � 38 � 38 ◮ So there are at most �� � � ��� � 3 + 2 + 4 other shapes. 3 4 5

Maximal families ( f = ⌊ N + 1 Example: N = 38, v = 7 v ⌋ = 5, d = v ( f + 1 ) − N = 4) � 38 � 38 ◮ There are at most � � + special shapes containing a 1 or 2. 1 2 ◮ Let the defect of a part x � 5 be f + 1 − x = 6 − x . � 38 � 38 � 38 ◮ The total defect in the other shapes is at most 3 � � � + 2 + . 3 4 5 ◮ Any shape has total defect at least d = 4. � 38 � 38 � 38 ◮ So there are at most �� � � ��� � 3 + 2 + 4 other shapes. 3 4 5 � 38 � 38 � 38 � 38 � 38 ◮ So at most � � �� � � ��� � + + 3 + 2 + 4 shapes in total. 1 2 3 4 5

Maximal families ( f = ⌊ N + 1 Example: N = 38, v = 7 v ⌋ = 5, d = v ( f + 1 ) − N = 4) � 38 � 38 ◮ There are at most � � + special shapes containing a 1 or 2. 1 2 ◮ Let the defect of a part x � 5 be f + 1 − x = 6 − x . � 38 � 38 � 38 ◮ The total defect in the other shapes is at most 3 � � � + 2 + . 3 4 5 ◮ Any shape has total defect at least d = 4. � 38 � 38 � 38 ◮ So there are at most �� � � ��� � 3 + 2 + 4 other shapes. 3 4 5 � 38 � 38 � 38 � 38 � 38 ◮ So at most � � �� � � ��� � + + 3 + 2 + 4 shapes in total. 1 2 3 4 5

Maximal families ( f = ⌊ N + 1 Example: N = 38, v = 7 v ⌋ = 5, d = v ( f + 1 ) − N = 4) � 38 � 38 ◮ There are at most � � + special shapes containing a 1 or 2. 1 2 ◮ Let the defect of a part x � 5 be f + 1 − x = 6 − x . � 38 � 38 � 38 ◮ The total defect in the other shapes is at most 3 � � � + 2 + . 3 4 5 ◮ Any shape has total defect at least d = 4. � 38 � 38 � 38 ◮ So there are at most �� � � ��� � 3 + 2 + 4 other shapes. 3 4 5 � 38 � 38 � 38 � 38 � 38 ◮ So at most � � �� � � ��� � + + 3 + 2 + 4 shapes in total. 1 2 3 4 5 � 38 � [ 1 , 6 , 6 , 6 , 6 , 6 , 7 ] × 1

Maximal families ( f = ⌊ N + 1 Example: N = 38, v = 7 v ⌋ = 5, d = v ( f + 1 ) − N = 4) � 38 � 38 ◮ There are at most � � + special shapes containing a 1 or 2. 1 2 ◮ Let the defect of a part x � 5 be f + 1 − x = 6 − x . � 38 � 38 � 38 ◮ The total defect in the other shapes is at most 3 � � � + 2 + . 3 4 5 ◮ Any shape has total defect at least d = 4. � 38 � 38 � 38 ◮ So there are at most �� � � ��� � 3 + 2 + 4 other shapes. 3 4 5 � 38 � 38 � 38 � 38 � 38 ◮ So at most � � �� � � ��� � + + 3 + 2 + 4 shapes in total. 1 2 3 4 5 � 38 � [ 1 , 6 , 6 , 6 , 6 , 6 , 7 ] × 1 � 38 � [ 2 , 6 , 6 , 6 , 6 , 6 , 6 ] × 2

Maximal families ( f = ⌊ N + 1 Example: N = 38, v = 7 v ⌋ = 5, d = v ( f + 1 ) − N = 4) � 38 � 38 ◮ There are at most � � + special shapes containing a 1 or 2. 1 2 ◮ Let the defect of a part x � 5 be f + 1 − x = 6 − x . � 38 � 38 � 38 ◮ The total defect in the other shapes is at most 3 � � � + 2 + . 3 4 5 ◮ Any shape has total defect at least d = 4. � 38 � 38 � 38 ◮ So there are at most �� � � ��� � 3 + 2 + 4 other shapes. 3 4 5 � 38 � 38 � 38 � 38 � 38 ◮ So at most � � �� � � ��� � + + 3 + 2 + 4 shapes in total. 1 2 3 4 5 � 38 � [ 1 , 6 , 6 , 6 , 6 , 6 , 7 ] × 1 � 38 � [ 2 , 6 , 6 , 6 , 6 , 6 , 6 ] × 2 � 38 � [ 3 , 5 , 6 , 6 , 6 , 6 , 6 ] × 3

Maximal families ( f = ⌊ N + 1 Example: N = 38, v = 7 v ⌋ = 5, d = v ( f + 1 ) − N = 4) � 38 � 38 ◮ There are at most � � + special shapes containing a 1 or 2. 1 2 ◮ Let the defect of a part x � 5 be f + 1 − x = 6 − x . � 38 � 38 � 38 ◮ The total defect in the other shapes is at most 3 � � � + 2 + . 3 4 5 ◮ Any shape has total defect at least d = 4. � 38 � 38 � 38 ◮ So there are at most �� � � ��� � 3 + 2 + 4 other shapes. 3 4 5 � 38 � 38 � 38 � 38 � 38 ◮ So at most � � �� � � ��� � + + 3 + 2 + 4 shapes in total. 1 2 3 4 5 � 38 � [ 1 , 6 , 6 , 6 , 6 , 6 , 7 ] × 1 � 38 � [ 2 , 6 , 6 , 6 , 6 , 6 , 6 ] × 2 � 38 � [ 3 , 5 , 6 , 6 , 6 , 6 , 6 ] × 3 � 38 � [ 4 , 5 , 5 , 6 , 6 , 6 , 6 ] × 4

Maximal families ( f = ⌊ N + 1 Example: N = 38, v = 7 v ⌋ = 5, d = v ( f + 1 ) − N = 4) � 38 � 38 ◮ There are at most � � + special shapes containing a 1 or 2. 1 2 ◮ Let the defect of a part x � 5 be f + 1 − x = 6 − x . � 38 � 38 � 38 ◮ The total defect in the other shapes is at most 3 � � � + 2 + . 3 4 5 ◮ Any shape has total defect at least d = 4. � 38 � 38 � 38 ◮ So there are at most �� � � ��� � 3 + 2 + 4 other shapes. 3 4 5 � 38 � 38 � 38 � 38 � 38 ◮ So at most � � �� � � ��� � + + 3 + 2 + 4 shapes in total. 1 2 3 4 5 � 38 � [ 1 , 6 , 6 , 6 , 6 , 6 , 7 ] × 1 � 38 � [ 2 , 6 , 6 , 6 , 6 , 6 , 6 ] × 2 � 38 � [ 3 , 5 , 6 , 6 , 6 , 6 , 6 ] × 3 � 38 � [ 4 , 5 , 5 , 6 , 6 , 6 , 6 ] × 4 ��� 38 � 38 � 38 � � ��� � [ 5 , 5 , 5 , 5 , 6 , 6 , 6 ] × − 2 − 4 5 4 3

Maximal families ( f = ⌊ N + 1 Example: N = 38, v = 7 v ⌋ = 5, d = v ( f + 1 ) − N = 4) � 38 � 38 ◮ There are at most � � + special shapes containing a 1 or 2. 1 2 ◮ Let the defect of a part x � 5 be f + 1 − x = 6 − x . � 38 � 38 � 38 ◮ The total defect in the other shapes is at most 3 � � � + 2 + . 3 4 5 ◮ Any shape has total defect at least d = 4. � 38 � 38 � 38 ◮ So there are at most �� � � ��� � 3 + 2 + 4 other shapes. 3 4 5 � 38 � 38 � 38 � 38 � 38 ◮ So at most � � �� � � ��� � + + 3 + 2 + 4 shapes in total. 1 2 3 4 5 � 38 � [ 1 , 6 , 6 , 6 , 6 , 6 , 7 ] × 1 � 38 � [ 2 , 6 , 6 , 6 , 6 , 6 , 6 ] × 2 � 38 � [ 3 , 5 , 6 , 6 , 6 , 6 , 6 ] × 3 � 38 � [ 4 , 5 , 5 , 6 , 6 , 6 , 6 ] × 4 ��� 38 � 38 � 38 � � ��� � [ 5 , 5 , 5 , 5 , 6 , 6 , 6 ] × − 2 − 4 5 4 3 � 38 � 38 � � Close to 5s, and fewer than 6s are used. 5 6

Maximal families

Maximal families The idea on the previous slide works unless N ≡ v − 1 ( mod v ) .

Maximal families The idea on the previous slide works unless N ≡ v − 1 ( mod v ) . ( f = ⌊ N + 1 Example: N = 41, v = 7 v ⌋ = 6, d = v ( f + 1 ) − N = 8)

Maximal families The idea on the previous slide works unless N ≡ v − 1 ( mod v ) . ( f = ⌊ N + 1 Example: N = 41, v = 7 v ⌋ = 6, d = v ( f + 1 ) − N = 8) � 41 � [ 1 , 6 , 6 , 7 , 7 , 7 , 7 ] × 1

Maximal families The idea on the previous slide works unless N ≡ v − 1 ( mod v ) . ( f = ⌊ N + 1 Example: N = 41, v = 7 v ⌋ = 6, d = v ( f + 1 ) − N = 8) � 41 � [ 1 , 6 , 6 , 7 , 7 , 7 , 7 ] × 1 � 41 � [ 2 , 6 , 6 , 6 , 7 , 7 , 7 ] × 2

Maximal families The idea on the previous slide works unless N ≡ v − 1 ( mod v ) . ( f = ⌊ N + 1 Example: N = 41, v = 7 v ⌋ = 6, d = v ( f + 1 ) − N = 8) � 41 � [ 1 , 6 , 6 , 7 , 7 , 7 , 7 ] × 1 � 41 � [ 2 , 6 , 6 , 6 , 7 , 7 , 7 ] × 2 � 41 � [ 3 , 6 , 6 , 6 , 6 , 7 , 7 ] × 3

Maximal families The idea on the previous slide works unless N ≡ v − 1 ( mod v ) . ( f = ⌊ N + 1 Example: N = 41, v = 7 v ⌋ = 6, d = v ( f + 1 ) − N = 8) � 41 � [ 1 , 6 , 6 , 7 , 7 , 7 , 7 ] × 1 � 41 � [ 2 , 6 , 6 , 6 , 7 , 7 , 7 ] × 2 � 41 � [ 3 , 6 , 6 , 6 , 6 , 7 , 7 ] × 3 � 41 � [ 4 , 6 , 6 , 6 , 6 , 6 , 7 ] × 4

Maximal families The idea on the previous slide works unless N ≡ v − 1 ( mod v ) . ( f = ⌊ N + 1 Example: N = 41, v = 7 v ⌋ = 6, d = v ( f + 1 ) − N = 8) � 41 � [ 1 , 6 , 6 , 7 , 7 , 7 , 7 ] × 1 � 41 � [ 2 , 6 , 6 , 6 , 7 , 7 , 7 ] × 2 � 41 � [ 3 , 6 , 6 , 6 , 6 , 7 , 7 ] × 3 � 41 � [ 4 , 6 , 6 , 6 , 6 , 6 , 7 ] × 4 � 41 � [ 5 , 6 , 6 , 6 , 6 , 6 , 6 ] × 5

Maximal families The idea on the previous slide works unless N ≡ v − 1 ( mod v ) . ( f = ⌊ N + 1 Example: N = 41, v = 7 v ⌋ = 6, d = v ( f + 1 ) − N = 8) � 41 � [ 1 , 6 , 6 , 7 , 7 , 7 , 7 ] × 1 � 41 � [ 2 , 6 , 6 , 6 , 7 , 7 , 7 ] × 2 � 41 � [ 3 , 6 , 6 , 6 , 6 , 7 , 7 ] × 3 � 41 � [ 4 , 6 , 6 , 6 , 6 , 6 , 7 ] × 4 � 41 � [ 5 , 6 , 6 , 6 , 6 , 6 , 6 ] × 5 Too many 6s.