Best Known (172−113, 172, s)-Nets in Base 3
(172−113, 172, 48)-Net over F3 — Constructive and digital
Digital (59, 172, 48)-net over F3, using
- t-expansion [i] based on digital (45, 172, 48)-net over F3, using
- net from sequence [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
- net from sequence [i] based on digital (45, 47)-sequence over F3, using
(172−113, 172, 64)-Net over F3 — Digital
Digital (59, 172, 64)-net over F3, using
- t-expansion [i] based on digital (49, 172, 64)-net over F3, using
- net from sequence [i] based on digital (49, 63)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 49 and N(F) ≥ 64, using
- net from sequence [i] based on digital (49, 63)-sequence over F3, using
(172−113, 172, 209)-Net over F3 — Upper bound on s (digital)
There is no digital (59, 172, 210)-net over F3, because
- 2 times m-reduction [i] would yield digital (59, 170, 210)-net over F3, but
- extracting embedded orthogonal array [i] would yield linear OA(3170, 210, F3, 111) (dual of [210, 40, 112]-code), but
- residual code [i] would yield OA(359, 98, S3, 37), but
- the linear programming bound shows that M ≥ 669 706575 926673 602832 294393 126699 440696 899104 765650 881469 270113 504934 632053 / 45175 106642 610955 544989 816374 596571 195177 978293 > 359 [i]
- residual code [i] would yield OA(359, 98, S3, 37), but
- extracting embedded orthogonal array [i] would yield linear OA(3170, 210, F3, 111) (dual of [210, 40, 112]-code), but
(172−113, 172, 259)-Net in Base 3 — Upper bound on s
There is no (59, 172, 260)-net in base 3, because
- 1 times m-reduction [i] would yield (59, 171, 260)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 4552 795636 440215 338239 977023 024765 815364 867687 353481 780714 476228 950938 074391 517505 > 3171 [i]