Best Known (81, 81+149, s)-Nets in Base 3
(81, 81+149, 56)-Net over F3 — Constructive and digital
Digital (81, 230, 56)-net over F3, using
- net from sequence [i] based on digital (81, 55)-sequence over F3, using
- base reduction for sequences [i] based on digital (13, 55)-sequence over F9, using
- s-reduction based on digital (13, 63)-sequence over F9, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 13 and N(F) ≥ 64, using
- s-reduction based on digital (13, 63)-sequence over F9, using
- base reduction for sequences [i] based on digital (13, 55)-sequence over F9, using
(81, 81+149, 84)-Net over F3 — Digital
Digital (81, 230, 84)-net over F3, using
- t-expansion [i] based on digital (71, 230, 84)-net over F3, using
- net from sequence [i] based on digital (71, 83)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 71 and N(F) ≥ 84, using
- net from sequence [i] based on digital (71, 83)-sequence over F3, using
(81, 81+149, 292)-Net over F3 — Upper bound on s (digital)
There is no digital (81, 230, 293)-net over F3, because
- 2 times m-reduction [i] would yield digital (81, 228, 293)-net over F3, but
- extracting embedded orthogonal array [i] would yield linear OA(3228, 293, F3, 147) (dual of [293, 65, 148]-code), but
- residual code [i] would yield OA(381, 145, S3, 49), but
- the linear programming bound shows that M ≥ 5454 903125 674302 038138 022709 253118 870603 680604 996309 484668 631548 163854 828465 899522 845238 627953 731491 227462 708997 548509 054594 512076 840977 032054 473727 036116 069732 583516 742090 980902 781249 742853 019393 892253 960097 248495 972677 421373 426141 628074 366647 / 12 085062 732503 338992 176763 549129 519718 479110 268997 150451 737221 775723 050593 147904 271161 753240 445212 100862 734331 956180 467283 411940 172483 857447 041584 506698 884680 282731 811631 103778 314212 762058 997686 304963 565740 > 381 [i]
- residual code [i] would yield OA(381, 145, S3, 49), but
- extracting embedded orthogonal array [i] would yield linear OA(3228, 293, F3, 147) (dual of [293, 65, 148]-code), but
(81, 81+149, 356)-Net in Base 3 — Upper bound on s
There is no (81, 230, 357)-net in base 3, because
- 1 times m-reduction [i] would yield (81, 229, 357)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 20 307546 163237 029922 235087 076289 674195 598422 323502 116361 615839 796602 417703 284786 926609 016053 910085 705440 935089 > 3229 [i]