Best Known (80, 80+135, s)-Nets in Base 3
(80, 80+135, 55)-Net over F3 — Constructive and digital
Digital (80, 215, 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+135, 84)-Net over F3 — Digital
Digital (80, 215, 84)-net over F3, using
- t-expansion [i] based on digital (71, 215, 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+135, 366)-Net over F3 — Upper bound on s (digital)
There is no digital (80, 215, 367)-net over F3, because
- extracting embedded orthogonal array [i] would yield linear OA(3215, 367, F3, 135) (dual of [367, 152, 136]-code), but
- residual code [i] would yield OA(380, 231, S3, 45), but
- 3 times truncation [i] would yield OA(377, 228, S3, 42), but
- the linear programming bound shows that M ≥ 5912 215514 524704 110906 945136 883000 212719 987869 884535 855049 744419 389403 700722 336619 158801 044684 / 1072 032675 663595 149019 556086 689047 982002 296477 532875 566121 > 377 [i]
- 3 times truncation [i] would yield OA(377, 228, S3, 42), but
- residual code [i] would yield OA(380, 231, S3, 45), but
(80, 80+135, 368)-Net in Base 3 — Upper bound on s
There is no (80, 215, 369)-net in base 3, because
- 1 times m-reduction [i] would yield (80, 214, 369)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 1 428234 495930 218687 920533 054878 099893 655467 181349 163163 728798 576177 833823 056222 735091 532361 599387 443035 > 3214 [i]