Best Known (80, 80+142, s)-Nets in Base 3
(80, 80+142, 55)-Net over F3 — Constructive and digital
Digital (80, 222, 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+142, 84)-Net over F3 — Digital
Digital (80, 222, 84)-net over F3, using
- t-expansion [i] based on digital (71, 222, 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+142, 347)-Net over F3 — Upper bound on s (digital)
There is no digital (80, 222, 348)-net over F3, because
- 1 times m-reduction [i] would yield digital (80, 221, 348)-net over F3, but
- extracting embedded orthogonal array [i] would yield linear OA(3221, 348, F3, 141) (dual of [348, 127, 142]-code), but
- residual code [i] would yield OA(380, 206, S3, 47), but
- 5 times truncation [i] would yield OA(375, 201, S3, 42), but
- the linear programming bound shows that M ≥ 70 125936 773107 976834 379244 843784 716640 335910 577502 309916 609029 015855 123273 912868 189563 572000 000000 / 109 901819 499502 271926 662008 620205 982068 132289 165070 372767 942049 > 375 [i]
- 5 times truncation [i] would yield OA(375, 201, S3, 42), but
- residual code [i] would yield OA(380, 206, S3, 47), but
- extracting embedded orthogonal array [i] would yield linear OA(3221, 348, F3, 141) (dual of [348, 127, 142]-code), but
(80, 80+142, 357)-Net in Base 3 — Upper bound on s
There is no (80, 222, 358)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 9724 697652 444235 665209 443635 702715 747470 695448 185605 452591 137997 178459 963176 601207 173788 297583 604775 934489 > 3222 [i]