Ann. Acad. Rom. Sci.

Ser. Math. Appl.

Vol. 18, No. 2/2026

ISSN 2066-6594

A SIMPLE CONSTRUCTION OF KIRKMAN

TRIPLE SYSTEMS OF ORDER 3^h∗

Antonio Causa^†

Leonardo Fragapane^‡

Elena Guardo^¶

Mario Gionfriddo^§

Communicated by G. Failla

DOI

10.56082/annalsarscimath.2026.2.225

Abstract

A Steiner triple system (STS) of order v is a 3-uniform hypergraph

with v vertices in which every 2-subset of vertices has degree 1. A

Kirkman triple system (KTS) is a resolvable Steiner triple system,

that is, a partition of the blocks of the triple system into classes which

are themselves partitions of the set of vertices into disjoint blocks. In

this paper we give a construction of KTS of order v = 3^hmuch simpler

and less technical than previously known constructions.

Keywords: Kirkman triple systems, resolvable designs.

MSC: 05B07, 05B10.

1 Introduction

A Steiner system S(h, k, v) is a pair Σ = (X, B), where X is a v-set and B

is a family of k-subsets of X such that every h-subset of X is contained in

^∗Accepted for publication on March 03, 2026

^†causa@dmi.unict.it, Dipartimento di Matematica e Informatica, University of Cata-

nia, Viale A. Doria, 6 - 95100 - Catania, Italy

^‡leonardo.fragapane@phd.unict.it, Dipartimento di Matematica e Informatica, Uni-

versity of Catania, Viale A. Doria, 6 - 95100 - Catania, Italy

^§gionfriddo@dmi.unict.it, Dipartimento di Matematica e Informatica, University of

Catania, Viale A. Doria, 6 - 95100 - Catania, Italy

^¶guardo@dmi.unict.it, Dipartimento di Matematica e Informatica, University of

Catania, Viale A. Doria, 6 - 95100 - Catania, Italy

225

A simple construction of KTS

226

exactly one member of B [4,6,10,11]. Using Hypergraph Theory terminology,

a Steiner system is a hypergraph Σ = (X, B) of order v, uniform of rank k,

such that every h-subset Y ⊆ X has degree d(Y ) = 1 [1,14 ]. Steiner systems

S(h, k, v) were deﬁned for the ﬁrst time by Woolhouse in 1844 [15], who

asked for which positive integers h, k, v an S(h, k, v) exists. This problem

remains unsolved in general until today, even if many partial results have

been given. A Steiner Triple System (STS) is a system S(2, 3, v), in 1847

T. Kirkman [9] and J. Steiner [13 ], independently, showed that an STS(v)

exists if and only if v ≡ 1 or 3 (mod 6).

Actually, there is an additional parameter to be considered, the so-called

index. A Steiner system S_λ(h, k, v) is a pair Σ = (X, B), where X is a v-set

and B is a family of k-subset of X such that every h-subset of X is contained

in exactly λ members of B. The ﬁrst deﬁnition of Steiner system was given

considering λ = 1

Other results have been determined by H. Hanani about the spectrum of

S(3, 4, v) and S(2, 4, v), respectively in 1960 [7] and in 1962 [8]. After the

solution on the spectrum of STS, the famous ﬁfteen schoolgirl problem was

posed. It is a problem proposed by Thomas Penyngton Kirkman in 1850

as Query VI in The Lady’s and Gentleman’s Diary (pag.48). The problem

states:

“Fifteen young ladies in a school walk out three abreast for seven days

in succession: it is required to arrange them daily so that no two shall walk

twice abreast.”

A solution to this problem is an example of a Kirkman triple system

(KTS) [10] which is a Steiner triple system having a parallelism, that is,

a partition of the blocks of the triple system into parallel classes which are

themselves partitions of the points into disjoint blocks. Such Steiner systems

that have a parallelism are also called resolvable and the partition is called

a resolution of the systems.

Soon after the problem about the ﬁfteen schoolgirls, in 1851 T. Kirkman

determined the spectrum of all the KTS proving that: a KTS exists if and

only if v ≡ 3 (mod 6).

Many constructions of Kirkman triple systems of all admissible orders

are already known in the literature, and this research still attracts a lot of

interest (see [2,3,5,12 ,16] just to cite some of them). In this paper we give

a construction of KTS having order v = 3^h, using the so-called method of

diﬀerences, which allows us to obtain a construction that is much simpler

and less technical than those previously available in the literature.

A. Causa, L. Fragapane, M. Gionfriddo, E. Guardo

227

2 Partition of diﬀerences

Let v = 3^h. Consider Z_vand deﬁne

ꢀ

ꢁ

v − 1

D_v= 1, 2, . . . ,

2

which is called set of diﬀerences in Z_v.

Let F = {F₁, F₂, . . . , F_r} be the partition of D_vsuch that:

∀ i = 1, . . . , r

a ∈ F_i⇐⇒ a is multiple of 3ⁱ⁻¹and a not multiple of 3ⁱ

Theorem 1. If F_i= {a_i,1, . . . , a_i,s}, for i = 1, . . . , r, and a_i,1= min_ja_i,j,

i

then:

1. a_i,1= 3ⁱ⁻¹

;

2. we deﬁne a_i,j+1as follows,

(

v−1

2

2a_i,j

if 2a_i,j

≤

a_i,j+1

=

;

v−1

v − 2a_i,j

if 2a_i,j

>

2

3. |F_i| = 3^h−i

;

4. |F| = h.

Proof.

1. By deﬁnition, 3ⁱ⁻¹is the minimum of F_i, then a_i,1= 3ⁱ⁻¹

.

2. The statement follows considering that v = 3^hand a_i,j+1is a multiple

of 3ⁱ⁻¹if and only if both 2a_i,jand v −2a_i,jare multiple of 3ⁱ⁻¹(same

for the other condition).

3. By deﬁnition, F_h= {3^h−1} and |F_h| = 1 = 3⁰.

• Consider F_h−1

.

There are nine multiples of 3^h−2in Z_v, namely 3^h−2, 2·3^h−2, . . . , 9·

3^h−2

:

– one of them is v;

– among the remaining eight, four belong to D_vwhile the other

four belong to Z_v\ D_v;

– among the four elements in D_v, there is 3^h−1∈ F_h.

Therefore, |F_h−1| = 3 = 3¹

A simple construction of KTS

228

• F_i, for 1 ≤ i ≤ h − 2.

There are 3^h−i+1multiples of 3ⁱ⁻¹in Z_v:

– one of them is v;

3^h−i+1−1

– among the remaining 3^h−i+1−1,

belong to D_vwhile

2

3^h−i+1−1

the other

belong to Z_v\ D_v;

elements in D_v, there are also the ones

2

– among the

2

which already belong to the sets F_k, for k > i.

Therefore,

h

3^h−i+1− 1

X

|F_i| =

−

|F_j| =

2

j=i+1

h

(3 − 1)(3^h−i+ ... + 1)

X

=

−

3^h−j

=

2

j=i+1

= (3^h−i+ ... + 1) − (3^h−i−1+ ... + 1) = 3^h−i

.

4. By deﬁnition, F_i= ∅ for i = 1, . . . , h while F_h+1= ∅.

Observe that a_h,2= v − 2a_h,1= 3^h− 2 · 3^h−1= 3^h−1= a_h,1.

3 Base blocks and a system S₃(2, 3, v)

From now on, consider Z_v= {0, 1, . . . , v − 1}. For every a ∈ D_v, deﬁne the

base block of a, B_a, as:

B_a= {v − a, 0, a}.

v−1

B_ais either associated with the triple of diﬀerences {a, a, 2a} if 2a ≤

or

2

v−1

to {a, a, v − 2a} if 2a >

.

If a = a_h,1, the diﬀerences²for the base block are {a, a, a}, making it the one

and only base block having a_h,1as a diﬀerence (all the others appear in two

of them).

For every a ∈ D_v, let Γ_abe the set of traslates of B_a,

Γ_a= {B_a,k= {v − a + k, k, a + k} ⊆ Z_v: a ∈ D_v, k = 0, . . . , v − 1}.

S

Theorem 2. If X = Z_vand B =

Γ_a, then Σ = (X, B) is an

a∈D_v

S₃(2, 3, v), for v = 3^h.

A. Causa, L. Fragapane, M. Gionfriddo, E. Guardo

229

Proof. It is immediate that Σ is a uniform hypergraph of rank 3, with order

equal to v = 3^h. It remains to prove that every pair of distinct elements of

X, let it be {x, y} with x < y, has degree equal to 3, d(x, y) = 3.

Let a be the diﬀerence between x and y, a = y−x or a = v−(y−x) based on

which one belongs to D_v; consider the base block B_a= B_a,0= {v − a, 0, a}:

• if a = y − x, B_a= {v + x − y, 0, y − x} ⊆ Z_v;

• if a = v − (y − x), B = {v − v + y − x, 0, v + x − y} =

{y − x, 0, v + x − y}^a⊆ Z_v.

Independently from the expression of a, the base block is {v+x−y, 0, y−x}:

assume that a = y − x ∈ D_v.

v

3

At ﬁrst, let a = : in Γ_athere are

B_a,x= {v + 2x − y, x, y} B_a,y= {x, y, 2y − x}.

In this case, v + 2x − y = 2y − x, hence B_a,x= B_a,y

.

Consider now another base block (depending from the value of a):

= {v − ^y−x, 0, ^y−x} ⊆ Z_v;

^a₂,0

a

2

• if a is even, B = B

2

• if a is odd, B^v−a= B

= {^v+y−x, 0, ^v−y+x} ⊆ Z_v.

^v−a,0

2

It is important to specify that all the operations are done in Z_v.

For each case, we are interested in a speciﬁc translate:

• if a is even, B_a^x+y= {v + x, ^x+y, y} = {x, ^x+y, y};

,

2

• if a is odd, B_v−a^v+y+x= {v + y, ^v+y+x, v + x} = {y, ^v+y+x, x}.

,

2

This proves that d(x, y) ≥ 3.

v

The last case is a =

= 3^h−1, that is the only element of F_h: the base block

3

v

= {2 · , 0, ^v₃}.

v

^v₃,1

is B = B

3

Consider now these three translates:

v

3

^v₃,x

^v₃,y

B

= {2 ·

+ x, x, y};

v

= {x, y,

+ y};

3

v

^v₃,y+ ^v₃

B

= {y, y + , x}.

3

A simple construction of KTS

230

This proves that d(x, y) ≥ 3 even in this case.

In order to prove that d(x, y) = 3, it would be suﬃcient to consider that the

diﬀerence a does not appear in any other base block.

However, we will conﬁrm this result calculating the cardinality of B =

S

Γ_a:

a∈D_v

X

v − 1

v(v − 1)

|B| =

|Γ_a| =

· v = 3 ·

,

2

6

a∈D_v

exactly the number of blocks of an S₃(2, 3, v).

4 Construction of KT S(3^h)

In the sequel, we use the following notation. Given a base block

B_a= {v − a, 0, a},

written in this order, we will say that the element 0 is the central vertex of

B_a, the element a is its right vertex and the element v − a is its left vertex.

For every i = 1, . . . , h, associate with F_ithe set U_i⊆ Z_vdeﬁned as follows:

F_h→ U_h= {0, 1, . . . , 3^h−1− 1}

=

{u : u ≡ 0, 1, . . . , 3^h−1− 1 (mod 3^h)}

F_h−1→ U_h−1= {u : u ≡ 0, 1, . . . , 3^h−2− 1 (mod 3^h−1)}

.

F_i→ U_i= {u : u ≡ 0, 1, . . . , 3ⁱ⁻¹− 1 (mod 3ⁱ)}

.

F₃→ U₃= {u : u ≡ 0, 1, . . . , 8 (mod 27)}

F₂→ U₂= {u : u ≡ 0, 1, 2 (mod 9)}

F₁→ U₁= {u : u ≡ 0 (mod 3)}

Each diﬀerence a ∈ D_vbelongs to a single set F_iand detects a base block

B_a= {v − a, 0, a}.

Associate with each i = 1, . . . , h and a ∈ F_ithe following set:

C_a,i= {B_a,u= {v − a + u, u, a + u} : u ∈ U_i}.

Observe that the total amount of these sets is

h

X

v − 1

|F_i| =

3^h−i

=

.

2

i=1

A. Causa, L. Fragapane, M. Gionfriddo, E. Guardo

231

S

Theorem 3. Let C_i=

C_a,i. If X = Z_vand B =

_i=1,2,...,hC_i, then

a∈F_i

Σ = (X, B) is a STS of order v = 3^h.

Proof. It is immediate that Σ is a uniform hypergraph of rank 3, with order

equals to v = 3^h. It remains to prove that every pair of distinct elements of

Z_v, let it be {x, y} with x < y, appears in exactly one block.

Let a be the diﬀerence between x and y, a = y − x or a = v − (y − x).

Based on which one belongs to D_v, consider the base block B_a= B_a,0

{v − y + x, 0, y − x}.

=

There exists a single i such that a ∈ F_i. Consider

U_i= {u : u ≡ 0, 1, . . . , 3ⁱ⁻¹− 1 (mod 3)ⁱ}.

• If x ∈ U_i, the pair {x, y} is contained in the block B_a,x= {v − y +

2x, x, y}.

• If y ∈ U_i, the pair {x, y} is contained in the block B_a,y= {x, y, 2y −x}.

• If both x, y ∈/ U_i, then they are not central elements of the block con-

taining the pair. This means that we have to look at the previous

diﬀerence of a, which is a/2.

^a₂,x+ ^a₂

If a is even, the pair {x, y} is contained in the block B

= {x, x +

^a₂, y}.

If a is odd, then it exists at least an α ∈ F_isuch that a = v−2α. It fol-

lows that the pair {x, y} is contained in the block B_α,y+α= {y, y+α, x}.

Therefore, it always exists at least a block containing any given pair of

vertices.

In order to prove the uniqueness of the block, we proceed by counting the

blocks of the entire system:

h

X

v

v · (v − 1)

|B| =

|F_i|·

= (3^h−1+... +3+1)·(3−1)·

= (3^h−1)·

=

,

3

6

i=1

which is exactly the number of blocks of an STS(v).

Now, it remains to prove that the family

Π = {C_a,i:, i = 1, . . . , h; a ∈ F_i}

is a resolution for the system Σ = (Z_v, B). This means that every class C_a,i

of Π is a parallel class: in other words a class of pairwise disjoint blocks. At

last, it will be proved that Σ is a KTS(3^h).

A simple construction of KTS

232

Theorem 4. The family Π = {C_a,i: i = 1, . . . , h;

a ∈ F_i} is a resolution

for the system Σ = (Z_v, B). Hence, Σ is a KTS(3^h).

Proof. Consider any class C ∈ Π: there exist an i = 1, . . . , h and an a ∈ F_i

such that:

:

B_a,u= {v − a + u, u, a + u} ∈ C_a,i− C,

Therefore, C contains all of the blocks of type:

{x − a, x, x + a}, ∀x ∈ U_i: a ∈ F_i.

u ∈ U_i.

Now, we prove that C is a parallel class for the system Σ = (Z_v, B). First

of all, consider that all the central vertices x of these blocks of C are all

diﬀerent among them and this implies that also all the right vertices are

diﬀerent amomg them and all the left vertices are diﬀerent among them. We

prove that also vertices of diﬀerent type are diﬀerent among them.

Let x, y ∈ U_i, x < y, a = y − x ∈ F_i. Then x + a is a right vertex and y − a

is a left vertex.

We prove that:

1. Every right vertex is diﬀerent from every central vertex.

Indeed, if x + a = y, then in the the same class associated with the

diﬀerence a they should be the blocks {x−a, x, x+a = y} and {y−a =

x, y, y + a}, both containing the pair {x, y}. It follows that: x + a = y.

2. Every left vertex is diﬀerent from every central vertex.

Indeed, if y = x − a, then in the the same class associated with the

diﬀerence a they should be the blocks {x − a = y, x, x + a} and {y −

a, y = x − a, y + a = x}, both containing the pair {x, y}. It follows

that: x − a = y.

3. Every left vertex is diﬀerent from every right vertex.

Indeed, if y − a = x + a [or x − a = y + a], then y = x + 2a, x = y − 2a

and in the same class associated with the diﬀerence 2a ∈ F_i, there

should be the blocks

{x − 2a, x, x + 2a = y} and {y − 2a = x, y, y + 2a},

both containing the pair {x, y}. Observe that if the successor of a in

F_iwere v − 2a instead of 2a, the same situation would occur.

Therefore, in every class C ∈ Π there are v/3 pairwise disjoint blocks and Π

is so a resolution of B.

This proves that the system is a KTS of order v = 3^h.

A. Causa, L. Fragapane, M. Gionfriddo, E. Guardo

233

5 Construction of KT S(3^h) of small order

1) KTS of order v = 9 = 3²

Let v = 9 = 3², h = 2.

Let X = Z₉, the set of vertices, and D₉= {1, 2, 3, 4}.

Diﬀerences and central vertices

i = 1

F₁= {1, 2, 4} −→ U₁= {(0) − (3) − (6)]}

Diﬀerence Triples T₁: (1, 1, 2), (2, 2, 4), (4, 4, 1)

i = 2

F₂= {3} −→ U₂= {(0, 1, 2)}

Diﬀerence Triples T₂: (3, 3, 3)

Base blocks and derived blocks

Blocks from the diﬀerences in (F₁, U₁):

8 · 0 · 1

2 · 3 · 4

5 · 6 · 7

7 · 0 · 2

1 · 3 · 5

4 · 6 · 8

5 · 0 · 4

8 · 3 · 7

2 · 6 · 1

Blocks from the diﬀerences in (F₂, U₂):

6 · 0 · 3

7 · 1 · 4

8 · 2 · 5

2) KT S of order v = 27 = 3³

Let v = 27 = 3³, h = 3. Let X = Z₂₇be the set of vertices, and D₂₇

=

{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13}

Diﬀerences and central vertices

i = 1

F₁= {1, 2, 4, 8, 11, 5, 10, 7, 13} −→ U₁= {(0) − (3) − (6) − (9) − (12) −

A simple construction of KTS

234

(15) − (18) − (21) − (24)}

Diﬀerence Triples T₁: (1, 1, 2), (2, 2, 4), (4, 4, 8), (8, 8, 11), (11, 11, 5), (5, 5, 10),

(10, 10, 7), (7, 7, 13), (13, 13, 1)

i = 2

F₂= {3, 6, 12} −→ U₂= {(0 − 1 − 2) − (9 − 10 − 11) − (18 − 19 − 20)}

Diﬀerence Triples T₂: (3, 3, 6), (6, 6, 12), (12, 12, 3)

i = 3

F₃= {9} −→ U₃= {(0 − 1 − 2 − 3 − 4 − 5 − 6 − 7 − 8)}

Diﬀerence Triples T₃: (9, 9, 9)

Base blocks and derived blocks

Blocks from the diﬀerences in (F₁, U₁):

26 · 0 · 1

25 · 0 · 2

23 · 0 · 4

19 · 0 · 8

22 · 3 · 11

25 · 6 · 14

1 · 9 · 17

16 · 0 · 11

19 · 3 · 14

22 · 6 · 17

25 · 9 · 20

1 · 12 · 23

4 · 15 · 26

7 · 18 · 2

2 · 3 · 4

1 · 3 · 5

26 · 3 · 7

5 · 6 · 7

4 · 6 · 8

2 · 6 · 10

8 · 9 · 10

7 · 9 · 11

5 · 9 · 13

11 · 12 · 13

14 · 15 · 16

17 · 18 · 19

20 · 21 · 22

23 · 24 · 25

22 · 0 · 5

25 · 3 · 8

10 · 12 · 14

13 · 15 · 17

16 · 18 · 20

19 · 21 · 23

22 · 24 · 26

17 · 0 · 10

20 · 3 · 13

23 · 6 · 16

26 · 9 · 19

2 · 12 · 22

5 · 15 · 25

8 · 18 · 1

8 · 12 · 16

11 · 15 · 19

14 · 18 · 22

17 · 21 · 25

20 · 24 · 1

20 · 0 · 7

23 · 3 · 10

26 · 6 · 13

2 · 9 · 16

4 · 12 · 20

7 · 15 · 23

10 · 18 · 26

13 · 21 · 2

20 · 24 · 5

14 · 0 · 13

17 · 3 · 16

20 · 6 · 19

23 · 9 · 22

26 · 12 · 25

2 · 15 · 1

10 · 21 · 5

13 · 24 · 8

1 · 6 · 11

4 · 9 · 14

7 · 12 · 17

10 · 15 · 20

13 · 18 · 23

16 · 21 · 26

19 · 24 · 2

5 · 12 · 19

8 · 15 · 22

11 · 18 · 25

14 · 21 · 1

17 · 24 · 4

5 · 18 · 4

11 · 21 · 4

14 · 24 · 7

8 · 21 · 7

11 · 24 · 10

A. Causa, L. Fragapane, M. Gionfriddo, E. Guardo

235

Blocks from the diﬀerences in (F₂, U₂):

24 · 0 · 3

25 · 1 · 4

21 · 0 · 6

22 · 1 · 7

15 · 0 · 12

16 · 1 · 13

17 · 2 · 14

24 · 9 · 21

25 · 10 · 22

26 · 11 · 23

6 · 18 · 3

26 · 2 · 5

23 · 2 · 8

6 · 9 · 12

3 · 9 · 15

7 · 10 · 13

8 · 11 · 14

15 · 18 · 21

16 · 19 · 22

17 · 20 · 23

4 · 10 · 16

5 · 11 · 17

12 · 18 · 24

13 · 19 · 25

14 · 20 · 26

7 · 19 · 4

8 · 20 · 5

Blocks from the diﬀerences in (F₃, U₃):

18 · 0 · 9

19 · 1 · 10

20 · 2 · 11

21 · 3 · 12

22 · 4 · 13

23 · 5 · 14

24 · 6 · 15

25 · 7 · 16

26 · 8 · 17

3) KT S of order v = 81 = 3⁴

Let v = 81 = 3⁴, h = 4.

Let X = Z₈₁be the set of vertices, and D₈₁= {1, 2, 3, 4, . . . , 38, 39, 40}

Diﬀerence triples and central vertices

i = 1

F₁−→

U₁with

F₁= {1, 2, 4, 8, 16, 32, 17, 34, 13, 26, 29, 23, 35, 11, 22, 37, 7, 14, 28, 25, 31, 19,

38, 5, 10, 20, 40}

and

U₁= {(0) − (3) − (6) − (9) − (12) − (15) − (18) − (21) − (24) − (27) − (30) −

(33) − (36)− (39) − (42) − (45) − (48) − (51) − (54) − (57) − (60) − (63) −

(66) − (69) − (72) − (75) − (78)}

Diﬀerences Triples T₁:

(1, 1, 2), (2, 2, 4), (4, 4, 8), (8, 8, 16), (16, 16, 32), (32, 32, 17), (17, 17, 34), (34, 34, 13),

A simple construction of KTS

236

(13, 13, 26), (26, 26, 29), (29, 29, 23), (23, 23, 35), (35, 35, 11), (11, 11, 22), (22, 22, 37),

(37, 37, 7), (7, 7, 14), (14, 14, 28), (28, 28, 25), (25, 25, 31), (31, 31, 19), (19, 19, 38),

(38, 38, 5), (5, 5, 10), (10, 10, 20), (20, 20, 40), (40, 40, 1)

i = 2

F₂−→

U₂with

F₂= {3, 6, 12, 15, 21, 24, 30, 33, 39}

and

U₂= {(0 − 1 − 2) − (9 − 10 − 11) − (18 − 19 − 20) − (27 − 28 − 29)−

(36−37−38)−(45−46−47), (54−55−56)−(63−64−65)−(72−73−74)}

Diﬀerences Triples T₂:

(3, 3, 6), (6, 6, 12), (12, 12, 24), (24, 24, 33), (33, 33, 15),

(15, 15, 30), (30, 30, 21), (21, 21, 39), (39, 39, 3)

i = 3

F₃−→

U₃with

F₃= {9, 18, 36}

and

U₃= {(0 − 1 − 2 − 3 − 4 − 5 − 6 − 7 − 8) − (27 − 28 − 29 − 30−

31 − 32 − 33 − 34 − 35), (54 − 55 − 56 − 57 − 58 − 59 − 60 − 61 − 62)}

Diﬀerences Triples T₃:

(9, 9, 18), (18, 18, 36), (36, 36, 9)

i = 4

F₄−→

U₄with

F₄= {27}

and

U₄= {(0 − 1 − 2 − 3 − 4 − 5 − 6 − 7 − 8 − 9 − 10 − 11 − 12 − 13 − 14 − 15 −

16 − 17 − 18 − 19 − 20 − 21 − 22 − 23 − 24 − 25 − 26)}

Diﬀerences Triples T₄:

(27, 27, 27)

A. Causa, L. Fragapane, M. Gionfriddo, E. Guardo

237

Base blocks and derived blocks

Blocks from the diﬀerences in (F₁, U₁):

80 · 0 · 1

2 · 3 · 4

79 · 0 · 2

1 · 3 · 5

77 · 0 · 4

80 · 3 · 7

· · · 41 · 0 · 40

· · ·

44 · 3 · 43

47 · 6 · 46

50 · 9 · 49

5 · 6 · 7

4 · 6 · 8

2 · 6 · 10

8 · 9 · 10

11 · 12 · 13

14 · 15 · 16

7 · 9 · 11

10 · 12 · 14

13 · 15 · 17

5 · 9 · 13

8 · 12 · 16

11 · 15 · 19

· · · 53 · 12 · 52

· · · 56 · 15 · 55

.

71 · 72 · 73

74 · 75 · 76

77 · 78 · 79

70 · 72 · 74

73 · 75 · 77

76 · 78 · 80

68 · 72 · 76

71 · 75 · 79

74 · 78 · 1

· · · 32 · 72 · 31

· · · 35 · 75 · 34

· · · 38 · 78 · 37

Blocks from the diﬀerences in (F₂, U₂):

78 · 0 · 3

79 · 1 · 4

80 · 2 · 5

6 · 9 · 12

7 · 10 · 13

8 · 11 · 14

75 · 0 · 6

76 · 1 · 7

77 · 2 · 8

3 · 9 · 15

4 · 10 · 16

5 · 11 · 17

69 · 0 · 12

70 · 1 · 13

71 · 2 · 14

78 · 9 · 21

79 · 10 · 22

80 · 11 · 23

· · ·

42 · 0 · 39

43 · 1 · 40

44 · 2 · 41

51 · 9 · 48

52 · 10 · 49

53 · 11 · 50

· · ·

.

69 · 72 · 75

70 · 73 · 76

71 · 74 · 77

66 · 72 · 78

67 · 73 · 79

68 · 74 · 80

60 · 72 · 3

61 · 73 · 4

62 · 74 · 5

· · ·

33 · 72 · 30

34 · 73 · 31

35 · 74 · 32

A simple construction of KTS

238

Blocks from the diﬀerences in (F₃, U₃):

72 · 0 · 9

63 · 0 · 18

45 · 0 · 36

73 · 1 · 10

64 · 1 · 19

46 · 1 · 37

.

80 · 8 · 17

18 · 27 · 36

19 · 28 · 37

20 · 29 · 38

71 · 8 · 26

9 · 27 · 45

10 · 28 · 46

11 · 29 · 47

53 · 8 · 44

72 · 27 · 63

73 · 28 · 64

74 · 29 · 65

.

26 · 35 · 44

45 · 54 · 63

46 · 55 · 64

17 · 35 · 53

36 · 54 · 72

37 · 55 · 73

80 · 35 · 71

18 · 54 · 9

19 · 55 · 10

.

53 · 62 · 71

44 · 62 · 80

26 · 62 · 17

Blocks from the diﬀerences in (F₄, U₄):

54 · 0 · 27

55 · 1 · 28

56 · 2 · 29

57 · 3 · 30

58 · 4 · 31

59 · 5 · 32

60 · 6 · 33

61 · 7 · 34

62 · 8 · 35

63 · 9 · 36

72 · 18 · 45

73 · 19 · 46

74 · 20 · 47

75 · 21 · 48

76 · 22 · 49

77 · 23 · 50

78 · 24 · 51

79 · 25 · 52

80 · 26 · 53

64 · 10 · 37

65 · 11 · 38

66 · 12 · 39

67 · 13 · 40

68 · 14 · 41

69 · 15 · 42

70 · 16 · 43

71 · 17 · 44

Acknowledgments. Causa, Gionfriddo and Guardo were partially sup-

ported by the Universita degli Studi di Catania,"PIACERI 2024/26" and by

GNSAGA Indam; Guardo has been supported by the project PRIN 2022,

"0-dimensional Schemes, Tensor Theory and Applications", funded by the

European Union Next Generation EU, Mission 4, Component 2 – CUP:

E53D23005670006.

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