TY - JOUR

T1 - Constructions of optimal packing designs

AU - Yin, Jianxing

AU - Assaf, Ahmed M.

PY - 1998

Y1 - 1998

N2 - Let v and k be positive integers. A (v, k, 1)-packing design is an ordered pair (V, B) where V is a v-set and B is a collection of k-subsets of V (called blocks) such that every 2-subset of V occurs in at most one block of B. The packing problem is mainly to determine the packing number P(k, v), that is, the maximum number of blocks in such a packing design. It is well known that P(k, v) ≤ ⌊v⌊(v - 1)/(k - 1)⌋/k⌋ = J(k, v) where ⌊x⌋ denotes the greatest integer y such that y ≤ x. A (v, k, 1)-packing design having J(k, v) blocks is said to be optimal. In this article, we develop some general constructions to obtain optimal packing designs. As an application, we show that P(5, v) = J(5, v) if v ≡ 7, 11 or 15 (mod 20), with the exception of v ∈ {11, 15} and the possible exception of v ∈ {27, 47, 51, 67, 87, 135, 187, 231, 251, 291}.

AB - Let v and k be positive integers. A (v, k, 1)-packing design is an ordered pair (V, B) where V is a v-set and B is a collection of k-subsets of V (called blocks) such that every 2-subset of V occurs in at most one block of B. The packing problem is mainly to determine the packing number P(k, v), that is, the maximum number of blocks in such a packing design. It is well known that P(k, v) ≤ ⌊v⌊(v - 1)/(k - 1)⌋/k⌋ = J(k, v) where ⌊x⌋ denotes the greatest integer y such that y ≤ x. A (v, k, 1)-packing design having J(k, v) blocks is said to be optimal. In this article, we develop some general constructions to obtain optimal packing designs. As an application, we show that P(5, v) = J(5, v) if v ≡ 7, 11 or 15 (mod 20), with the exception of v ∈ {11, 15} and the possible exception of v ∈ {27, 47, 51, 67, 87, 135, 187, 231, 251, 291}.

KW - Construction

KW - Optimal packing

KW - Packing number

UR - http://www.scopus.com/inward/record.url?scp=0041692267&partnerID=8YFLogxK

U2 - 10.1002/(SICI)1520-6610(1998)6:4<245::AID-JCD3>3.0.CO;2-F

DO - 10.1002/(SICI)1520-6610(1998)6:4<245::AID-JCD3>3.0.CO;2-F

M3 - Article

AN - SCOPUS:0041692267

VL - 6

SP - 245

EP - 260

JO - Journal of Combinatorial Designs

JF - Journal of Combinatorial Designs

SN - 1063-8539

IS - 4

ER -