Best Known (66, 208, s)-Nets in Base 3
(66, 208, 48)-Net over F3 — Constructive and digital
Digital (66, 208, 48)-net over F3, using
- t-expansion [i] based on digital (45, 208, 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
(66, 208, 64)-Net over F3 — Digital
Digital (66, 208, 64)-net over F3, using
- t-expansion [i] based on digital (49, 208, 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
(66, 208, 206)-Net over F3 — Upper bound on s (digital)
There is no digital (66, 208, 207)-net over F3, because
- 7 times m-reduction [i] would yield digital (66, 201, 207)-net over F3, but
- extracting embedded orthogonal array [i] would yield linear OA(3201, 207, F3, 135) (dual of [207, 6, 136]-code), but
- residual code [i] would yield linear OA(366, 71, F3, 45) (dual of [71, 5, 46]-code), but
- “HW1†bound on codes from Brouwer’s database [i]
- residual code [i] would yield linear OA(366, 71, F3, 45) (dual of [71, 5, 46]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(3201, 207, F3, 135) (dual of [207, 6, 136]-code), but
(66, 208, 211)-Net in Base 3 — Upper bound on s
There is no (66, 208, 212)-net in base 3, because
- 1 times m-reduction [i] would yield (66, 207, 212)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3207, 212, S3, 141), but
- the (dual) Plotkin bound shows that M ≥ 47052 721287 394587 764057 094854 672253 553918 218437 190874 778408 030747 195017 485692 977810 906266 281547 645147 / 71 > 3207 [i]
- extracting embedded orthogonal array [i] would yield OA(3207, 212, S3, 141), but