Best Known (239−176, 239, s)-Nets in Base 3
(239−176, 239, 48)-Net over F3 — Constructive and digital
Digital (63, 239, 48)-net over F3, using
- t-expansion [i] based on digital (45, 239, 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
(239−176, 239, 64)-Net over F3 — Digital
Digital (63, 239, 64)-net over F3, using
- t-expansion [i] based on digital (49, 239, 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
(239−176, 239, 198)-Net over F3 — Upper bound on s (digital)
There is no digital (63, 239, 199)-net over F3, because
- 47 times m-reduction [i] would yield digital (63, 192, 199)-net over F3, but
- extracting embedded orthogonal array [i] would yield linear OA(3192, 199, F3, 129) (dual of [199, 7, 130]-code), but
- residual code [i] would yield linear OA(363, 69, F3, 43) (dual of [69, 6, 44]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(3192, 199, F3, 129) (dual of [199, 7, 130]-code), but
(239−176, 239, 202)-Net in Base 3 — Upper bound on s
There is no (63, 239, 203)-net in base 3, because
- 41 times m-reduction [i] would yield (63, 198, 203)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3198, 203, S3, 135), but
- the (dual) Plotkin bound shows that M ≥ 2 390525 899882 872924 049031 898322 016641 463101 073880 550463 771174 655651 832418 111719 646949 462291 396009 / 68 > 3198 [i]
- extracting embedded orthogonal array [i] would yield OA(3198, 203, S3, 135), but