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 -