Best Known (241−155, 241, s)-Nets in Base 3
(241−155, 241, 61)-Net over F3 — Constructive and digital
Digital (86, 241, 61)-net over F3, using
- net from sequence [i] based on digital (86, 60)-sequence over F3, using
- base reduction for sequences [i] based on digital (13, 60)-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, 60)-sequence over F9, using
(241−155, 241, 84)-Net over F3 — Digital
Digital (86, 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
(241−155, 241, 374)-Net over F3 — Upper bound on s (digital)
There is no digital (86, 241, 375)-net over F3, because
- 2 times m-reduction [i] would yield digital (86, 239, 375)-net over F3, but
- extracting embedded orthogonal array [i] would yield linear OA(3239, 375, F3, 153) (dual of [375, 136, 154]-code), but
- residual code [i] would yield linear OA(386, 221, F3, 51) (dual of [221, 135, 52]-code), but
- 1 times truncation [i] would yield linear OA(385, 220, F3, 50) (dual of [220, 135, 51]-code), but
- the Johnson bound shows that N ≤ 22905 993227 448745 319194 488151 012424 723463 351500 755461 393112 800141 < 3135 [i]
- 1 times truncation [i] would yield linear OA(385, 220, F3, 50) (dual of [220, 135, 51]-code), but
- residual code [i] would yield linear OA(386, 221, F3, 51) (dual of [221, 135, 52]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(3239, 375, F3, 153) (dual of [375, 136, 154]-code), but
(241−155, 241, 381)-Net in Base 3 — Upper bound on s
There is no (86, 241, 382)-net in base 3, because
- 1 times m-reduction [i] would yield (86, 240, 382)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 3 796581 915971 615608 596523 149700 544706 311875 331184 992212 526237 337567 117588 488786 634024 713853 923480 188525 480032 168837 > 3240 [i]