Best Known (69, 69+∞, s)-Nets in Base 3
(69, 69+∞, 48)-Net over F3 — Constructive and digital
Digital (69, m, 48)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (69, 47)-sequence over F3, using
- t-expansion [i] based on digital (45, 47)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 45 and N(F) ≥ 48, using
- t-expansion [i] based on digital (45, 47)-sequence over F3, using
(69, 69+∞, 82)-Net over F3 — Digital
Digital (69, m, 82)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (69, 81)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 69 and N(F) ≥ 82, using
(69, 69+∞, 150)-Net in Base 3 — Upper bound on s
There is no (69, m, 151)-net in base 3 for arbitrarily large m, because
- m-reduction [i] would yield (69, 749, 151)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(3749, 151, S3, 5, 680), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 667909 510910 747021 953101 680043 594801 550328 863016 452492 339068 595406 821392 774310 475266 736188 979242 424321 746377 347804 845066 965098 509018 730642 033222 401425 061216 185186 444237 628416 828575 443400 834150 382233 335585 148202 868122 214831 685032 307533 984901 644631 483536 467330 211079 216888 323675 031122 270136 926330 562499 884771 544486 209159 351946 810021 008743 416509 250744 909371 031987 / 227 > 3749 [i]
- extracting embedded OOA [i] would yield OOA(3749, 151, S3, 5, 680), but