Best Known (80, 80+149, s)-Nets in Base 3
(80, 80+149, 55)-Net over F3 — Constructive and digital
Digital (80, 229, 55)-net over F3, using
- net from sequence [i] based on digital (80, 54)-sequence over F3, using
- base reduction for sequences [i] based on digital (13, 54)-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, 54)-sequence over F9, using
(80, 80+149, 84)-Net over F3 — Digital
Digital (80, 229, 84)-net over F3, using
- t-expansion [i] based on digital (71, 229, 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
(80, 80+149, 280)-Net over F3 — Upper bound on s (digital)
There is no digital (80, 229, 281)-net over F3, because
- 2 times m-reduction [i] would yield digital (80, 227, 281)-net over F3, but
- extracting embedded orthogonal array [i] would yield linear OA(3227, 281, F3, 147) (dual of [281, 54, 148]-code), but
- residual code [i] would yield OA(380, 133, S3, 49), but
- the linear programming bound shows that M ≥ 21223 654940 747716 870153 549604 409912 091138 635334 644884 306688 767122 427377 344322 374868 328420 535820 061178 192292 178417 467677 / 142 600278 894934 337416 159250 039930 631375 413133 442674 819708 793351 208507 551154 176000 > 380 [i]
- residual code [i] would yield OA(380, 133, S3, 49), but
- extracting embedded orthogonal array [i] would yield linear OA(3227, 281, F3, 147) (dual of [281, 54, 148]-code), but
(80, 80+149, 350)-Net in Base 3 — Upper bound on s
There is no (80, 229, 351)-net in base 3, because
- 1 times m-reduction [i] would yield (80, 228, 351)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 7 023016 105062 658852 723389 247436 275160 827680 732197 869238 712195 091720 570933 656062 342645 330801 038648 212986 804781 > 3228 [i]