Best Known (84, 84+166, s)-Nets in Base 3
(84, 84+166, 59)-Net over F3 — Constructive and digital
Digital (84, 250, 59)-net over F3, using
- net from sequence [i] based on digital (84, 58)-sequence over F3, using
- base reduction for sequences [i] based on digital (13, 58)-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, 58)-sequence over F9, using
(84, 84+166, 84)-Net over F3 — Digital
Digital (84, 250, 84)-net over F3, using
- t-expansion [i] based on digital (71, 250, 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
(84, 84+166, 264)-Net over F3 — Upper bound on s (digital)
There is no digital (84, 250, 265)-net over F3, because
- 1 times m-reduction [i] would yield digital (84, 249, 265)-net over F3, but
- extracting embedded orthogonal array [i] would yield linear OA(3249, 265, F3, 165) (dual of [265, 16, 166]-code), but
- residual code [i] would yield OA(384, 99, S3, 55), but
- the linear programming bound shows that M ≥ 4 602106 814071 993271 061293 737170 927222 039075 353067 / 326 668160 > 384 [i]
- residual code [i] would yield OA(384, 99, S3, 55), but
- extracting embedded orthogonal array [i] would yield linear OA(3249, 265, F3, 165) (dual of [265, 16, 166]-code), but
(84, 84+166, 357)-Net in Base 3 — Upper bound on s
There is no (84, 250, 358)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 223561 617269 563835 876730 634913 362520 604294 325161 835789 419668 366956 657529 870888 413264 098764 416513 069502 934201 304652 154945 > 3250 [i]