Some more 5-GDDs, 4-frames and 4-RGDDs

Wei, Hengjia; Ge, Gennian

doi:10.1016/j.disc.2014.07.014

Cited by 8 publications

(14 citation statements)

References 30 publications

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“…Since

v \equiv 10 (normalmod 20)

there exists a 5‐GDD of type

v^{2 t + 1}

for

t \geq 2

. This follows from [14, Theorem 2.25] when

t \neq 11

; otherwise the 5‐GDD of type

v^{23}

is constructed by Lemma 4.2 using the 5‐GDD of type

1 0^{23}

from Lemma 4.1.…”

Section: Block Sizementioning

confidence: 96%

“…Incidentally, the designs of Lemma 4.1 eliminate many possible exceptions from Theorem 2.25 of [14], including all those of type

{MathClass-open(10 α MathClass-close)}^{23}

. The next two lemmas are special cases of well‐known GDD constructions.…”

Section: Block Sizementioning

confidence: 98%

“…Proof Let

v = 4 r + 6

and note that

v \geq 86

. In each case we use Theorem 1.1 together with an appropriate 5‐GDD of type

v^{u}

for the existence of which we refer to [14, Theorem 2.25] unless otherwise stated. Since

v \equiv 6 (normalmod 20)

there exists a 5‐GDD of type

v^{10 t + 1}

for

t \geq 0

. Since

v \equiv 6 (normalmod 20)

there exists a 5‐GDD of type

v^{10 t + 5}

for

t \geq 0

except possibly when

v \leq 4886, gcd (v ∕ 2, 30) = 1

and

t = 1

, that is, when

r \leq 1220, r ≢ 0 (normalmod 3)

and

t = 1

. Since

v \equiv 10 (normalmod 20)

there exists a 5‐GDD of type

v^{2 t + 1}

for

t \geq 2

.…”

Section: Block Sizementioning

confidence: 99%

“…Proof The proof is similar to that of Lemma 4.6. There exist 5‐GDDs of the following types:

1 0^{11}

and

2 0^{11}

from [14, Theorem 2.25];

1 0^{10} 1 8^{1}

and

2 0^{10} 3 6^{1}

from Lemma 4.1;

1 0^{10} 3 0^{1}

and

2 0^{10} 6 0^{1}

by Lemma 4.3 from 4‐RGDDs of types

1 0^{10}

and

2 0^{10}

[14, Theorem 4.21]. Except for

g \in {30, 60}

we use Lemma 4.2 to construct 5‐GDDs of types

g^{10} {MathClass-open(g d ∕ 5 MathClass-close)}^{1}

for

d \in {5, 9, 15}

from the corresponding 5‐GDDs of types

1 0^{10} {MathClass-open(2 d MathClass-close)}^{1}

and

2 0^{10} {MathClass-open(4 d MathClass-close)}^{1}

.…”

Section: Block Sizementioning

confidence: 99%

“…Thus a

PENT (5, 2131)

is constructed from existing pentagonal geometries with no opposite line pairs, 40 copies of

PENT (5, 40)

and one of

PENT (5, 471)

. The

PENT (5, 471)

is obtained by Theorem 4.3 from 15 copies of

PENT (5, 30)

and a 5‐GDD of type

12 6^{15}

, which exists by [14, Theorem 2.25].…”

Section: Block Sizementioning

confidence: 99%

See 4 more Smart Citations

Pentagonal geometries with block sizes 3, 4, and 5

Forbes

2021

J of Combinatorial Designs

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A pentagonal geometry PENT(k,r) is a partial linear space, where every line, or block, is incident with k points, every point is incident with r lines, and for each point x, there is a line incident with precisely those points that are not collinear with x. An opposite line pair in a pentagonal geometry consists of two parallel lines such that each point on one of the lines is not collinear with precisely those points on the other line. We give a direct construction for an infinite sequence of pentagonal geometries with block size 3 and connected deficiency graphs. Also we present 39 new pentagonal geometries with block size 4 and five with block size 5, all with connected deficiency graphs. Consequentially we determine the existence spectrum up to a few possible exceptions for PENT(4,r) that do not contain opposite line pairs and for PENT(4,r) with one opposite line pair. More generally, given j we show that there exists a PENT(4,r) with j opposite line pairs for all sufficiently large admissible r. Using some new group divisible designs with block size 5 (including types 235,271, and 1023) we significantly extend the known existence spectrum for PENT(5,r).

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“…Since

v \equiv 10 (normalmod 20)

there exists a 5‐GDD of type

v^{2 t + 1}

for

t \geq 2

. This follows from [14, Theorem 2.25] when

t \neq 11

; otherwise the 5‐GDD of type

v^{23}

is constructed by Lemma 4.2 using the 5‐GDD of type

1 0^{23}

from Lemma 4.1.…”

Section: Block Sizementioning

confidence: 96%

“…Incidentally, the designs of Lemma 4.1 eliminate many possible exceptions from Theorem 2.25 of [14], including all those of type

{MathClass-open(10 α MathClass-close)}^{23}

. The next two lemmas are special cases of well‐known GDD constructions.…”

Section: Block Sizementioning

confidence: 98%

“…Proof Let

v = 4 r + 6

and note that

v \geq 86

. In each case we use Theorem 1.1 together with an appropriate 5‐GDD of type

v^{u}

for the existence of which we refer to [14, Theorem 2.25] unless otherwise stated. Since

v \equiv 6 (normalmod 20)

there exists a 5‐GDD of type

v^{10 t + 1}

for

t \geq 0

. Since

v \equiv 6 (normalmod 20)

there exists a 5‐GDD of type

v^{10 t + 5}

for

t \geq 0

except possibly when

v \leq 4886, gcd (v ∕ 2, 30) = 1

and

t = 1

, that is, when

r \leq 1220, r ≢ 0 (normalmod 3)

and

t = 1

. Since

v \equiv 10 (normalmod 20)

there exists a 5‐GDD of type

v^{2 t + 1}

for

t \geq 2

.…”

Section: Block Sizementioning

confidence: 99%

“…Proof The proof is similar to that of Lemma 4.6. There exist 5‐GDDs of the following types:

1 0^{11}

and

2 0^{11}

from [14, Theorem 2.25];

1 0^{10} 1 8^{1}

and

2 0^{10} 3 6^{1}

from Lemma 4.1;

1 0^{10} 3 0^{1}

and

2 0^{10} 6 0^{1}

by Lemma 4.3 from 4‐RGDDs of types

1 0^{10}

and

2 0^{10}

[14, Theorem 4.21]. Except for

g \in {30, 60}

we use Lemma 4.2 to construct 5‐GDDs of types

g^{10} {MathClass-open(g d ∕ 5 MathClass-close)}^{1}

for

d \in {5, 9, 15}

from the corresponding 5‐GDDs of types

1 0^{10} {MathClass-open(2 d MathClass-close)}^{1}

and

2 0^{10} {MathClass-open(4 d MathClass-close)}^{1}

.…”

Section: Block Sizementioning

confidence: 99%

“…Thus a

PENT (5, 2131)

is constructed from existing pentagonal geometries with no opposite line pairs, 40 copies of

PENT (5, 40)

and one of

PENT (5, 471)

. The

PENT (5, 471)

is obtained by Theorem 4.3 from 15 copies of

PENT (5, 30)

and a 5‐GDD of type

12 6^{15}

, which exists by [14, Theorem 2.25].…”

Section: Block Sizementioning

confidence: 99%

See 3 more Smart Citations

Pentagonal geometries with block sizes 3, 4, and 5

Forbes

2021

J of Combinatorial Designs

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Generalized Pentagonal Geometries-II

Forbes,

Rutherford

2024

La Matematica

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Some new resolvable GDDs with k=4 and doubly resolvable GDDs with k=3

Abel

Wang

2015

Discrete Mathematics

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a b s t r a c tA doubly resolvable packing design with block size k, index λ, replication number r, and v elements is called a generalized Kirkman square and denoted by GKS k (v; 1, λ; r). Existence of GKS 3 (4u; 1, 1; 2(u −1))s and GKS 3 (6u; 1, 1; 3(u −1))s is implied by existence of doubly resolvable group divisible designs with block size 3, index 1, and types 4 u and 6 u (i.e., (3, 1)-DRGDDs of types 4 u and 6 u ). In this paper, we establish the spectra of (3, 1)-DRGDDs of types 4 u and 6 u with 15 and 31 possible exceptions, respectively. As applications, we get some new classes of permutation codes and doubly constant weight codes. We also construct 5 new resolvable GDDs with block size 4 and index 1.

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Some more 5-GDDs, 4-frames and 4-RGDDs

Cited by 8 publications

References 30 publications

Pentagonal geometries with block sizes 3, 4, and 5

Pentagonal geometries with block sizes 3, 4, and 5

Generalized Pentagonal Geometries-II

Some new resolvable GDDs with k=4 and doubly resolvable GDDs with k=3

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