Best Known (50, 50+89, s)-Nets in Base 3
(50, 50+89, 48)-Net over F3 — Constructive and digital
Digital (50, 139, 48)-net over F3, using
- t-expansion [i] based on digital (45, 139, 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
(50, 50+89, 64)-Net over F3 — Digital
Digital (50, 139, 64)-net over F3, using
- t-expansion [i] based on digital (49, 139, 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
(50, 50+89, 222)-Net over F3 — Upper bound on s (digital)
There is no digital (50, 139, 223)-net over F3, because
- 2 times m-reduction [i] would yield digital (50, 137, 223)-net over F3, but
- extracting embedded orthogonal array [i] would yield linear OA(3137, 223, F3, 87) (dual of [223, 86, 88]-code), but
- residual code [i] would yield OA(350, 135, S3, 29), but
- 1 times truncation [i] would yield OA(349, 134, S3, 28), but
- the linear programming bound shows that M ≥ 430073 037911 139946 191996 083081 250824 628251 158946 484375 / 1 673823 880993 304164 747289 137687 > 349 [i]
- 1 times truncation [i] would yield OA(349, 134, S3, 28), but
- residual code [i] would yield OA(350, 135, S3, 29), but
- extracting embedded orthogonal array [i] would yield linear OA(3137, 223, F3, 87) (dual of [223, 86, 88]-code), but
(50, 50+89, 229)-Net in Base 3 — Upper bound on s
There is no (50, 139, 230)-net in base 3, because
- 1 times m-reduction [i] would yield (50, 138, 230)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 721887 702938 940292 356824 474902 631186 573035 656661 945159 675343 764377 > 3138 [i]