Best Known (208−130, 208, s)-Nets in Base 3
(208−130, 208, 53)-Net over F3 — Constructive and digital
Digital (78, 208, 53)-net over F3, using
- net from sequence [i] based on digital (78, 52)-sequence over F3, using
- base reduction for sequences [i] based on digital (13, 52)-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, 52)-sequence over F9, using
(208−130, 208, 84)-Net over F3 — Digital
Digital (78, 208, 84)-net over F3, using
- t-expansion [i] based on digital (71, 208, 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
(208−130, 208, 358)-Net over F3 — Upper bound on s (digital)
There is no digital (78, 208, 359)-net over F3, because
- 1 times m-reduction [i] would yield digital (78, 207, 359)-net over F3, but
- extracting embedded orthogonal array [i] would yield linear OA(3207, 359, F3, 129) (dual of [359, 152, 130]-code), but
- residual code [i] would yield OA(378, 229, S3, 43), but
- 1 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]
- 1 times truncation [i] would yield OA(377, 228, S3, 42), but
- residual code [i] would yield OA(378, 229, S3, 43), but
- extracting embedded orthogonal array [i] would yield linear OA(3207, 359, F3, 129) (dual of [359, 152, 130]-code), but
(208−130, 208, 360)-Net in Base 3 — Upper bound on s
There is no (78, 208, 361)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 1888 532195 728468 315475 851927 722873 920158 718562 516415 352674 201796 797299 052349 189787 519619 507157 454291 > 3208 [i]