Best Known (61, 170, s)-Nets in Base 3
(61, 170, 48)-Net over F3 — Constructive and digital
Digital (61, 170, 48)-net over F3, using
- t-expansion [i] based on digital (45, 170, 48)-net over F3, using
- net from sequence [i] based on digital (45, 47)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 45 and N(F) ≥ 48, using
- net from sequence [i] based on digital (45, 47)-sequence over F3, using
(61, 170, 64)-Net over F3 — Digital
Digital (61, 170, 64)-net over F3, using
- t-expansion [i] based on digital (49, 170, 64)-net over F3, using
- net from sequence [i] based on digital (49, 63)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 49 and N(F) ≥ 64, using
- net from sequence [i] based on digital (49, 63)-sequence over F3, using
(61, 170, 250)-Net over F3 — Upper bound on s (digital)
There is no digital (61, 170, 251)-net over F3, because
- 1 times m-reduction [i] would yield digital (61, 169, 251)-net over F3, but
- extracting embedded orthogonal array [i] would yield linear OA(3169, 251, F3, 108) (dual of [251, 82, 109]-code), but
- residual code [i] would yield OA(361, 142, S3, 36), but
- the linear programming bound shows that M ≥ 110057 996992 264008 335718 129149 812789 167410 382926 297281 137489 561817 954781 539797 095330 446330 796321 986335 249770 043338 391672 954660 756429 831103 370683 368332 400198 682368 203873 342849 181601 932753 536134 620000 / 801375 596792 772331 886384 195812 844385 927415 230504 801683 142250 629380 125033 284694 411117 301101 934018 200100 135754 437215 084825 133809 831804 737408 596792 027147 668977 826191 688627 > 361 [i]
- residual code [i] would yield OA(361, 142, S3, 36), but
- extracting embedded orthogonal array [i] would yield linear OA(3169, 251, F3, 108) (dual of [251, 82, 109]-code), but
(61, 170, 276)-Net in Base 3 — Upper bound on s
There is no (61, 170, 277)-net in base 3, because
- 1 times m-reduction [i] would yield (61, 169, 277)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 488 105500 332943 304248 675317 310312 850974 364101 753720 539507 809329 299697 448452 873377 > 3169 [i]