Best Known (241−154, 241, s)-Nets in Base 3
(241−154, 241, 62)-Net over F3 — Constructive and digital
Digital (87, 241, 62)-net over F3, using
- net from sequence [i] based on digital (87, 61)-sequence over F3, using
- base reduction for sequences [i] based on digital (13, 61)-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, 61)-sequence over F9, using
(241−154, 241, 84)-Net over F3 — Digital
Digital (87, 241, 84)-net over F3, using
- t-expansion [i] based on digital (71, 241, 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
(241−154, 241, 383)-Net over F3 — Upper bound on s (digital)
There is no digital (87, 241, 384)-net over F3, because
- 1 times m-reduction [i] would yield digital (87, 240, 384)-net over F3, but
- extracting embedded orthogonal array [i] would yield linear OA(3240, 384, F3, 153) (dual of [384, 144, 154]-code), but
- residual code [i] would yield linear OA(387, 230, F3, 51) (dual of [230, 143, 52]-code), but
- 1 times truncation [i] would yield linear OA(386, 229, F3, 50) (dual of [229, 143, 51]-code), but
- the Johnson bound shows that N ≤ 153 153734 778487 159186 543991 832667 435898 157821 534223 513753 575995 914628 < 3143 [i]
- 1 times truncation [i] would yield linear OA(386, 229, F3, 50) (dual of [229, 143, 51]-code), but
- residual code [i] would yield linear OA(387, 230, F3, 51) (dual of [230, 143, 52]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(3240, 384, F3, 153) (dual of [384, 144, 154]-code), but
(241−154, 241, 387)-Net in Base 3 — Upper bound on s
There is no (87, 241, 388)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 10 548795 724917 665078 015330 713411 563719 848244 448696 089036 441285 767779 802534 685907 262067 325852 624630 599619 141837 786729 > 3241 [i]