Best Known (60, 60+107, s)-Nets in Base 3
(60, 60+107, 48)-Net over F3 — Constructive and digital
Digital (60, 167, 48)-net over F3, using
- t-expansion [i] based on digital (45, 167, 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
(60, 60+107, 64)-Net over F3 — Digital
Digital (60, 167, 64)-net over F3, using
- t-expansion [i] based on digital (49, 167, 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
(60, 60+107, 253)-Net over F3 — Upper bound on s (digital)
There is no digital (60, 167, 254)-net over F3, because
- 2 times m-reduction [i] would yield digital (60, 165, 254)-net over F3, but
- extracting embedded orthogonal array [i] would yield linear OA(3165, 254, F3, 105) (dual of [254, 89, 106]-code), but
- residual code [i] would yield OA(360, 148, S3, 35), but
- the linear programming bound shows that M ≥ 32736 674181 247111 523198 040277 032547 478736 312432 761773 182980 156745 054076 913794 735387 713391 860471 994046 617776 080101 939907 837612 473720 955089 815789 352300 892160 / 691974 074846 423845 929231 601971 384016 174717 370524 100041 408052 710499 582060 732735 640083 715550 709077 494269 293566 444659 318525 072647 > 360 [i]
- residual code [i] would yield OA(360, 148, S3, 35), but
- extracting embedded orthogonal array [i] would yield linear OA(3165, 254, F3, 105) (dual of [254, 89, 106]-code), but
(60, 60+107, 272)-Net in Base 3 — Upper bound on s
There is no (60, 167, 273)-net in base 3, because
- 1 times m-reduction [i] would yield (60, 166, 273)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 18 025304 411806 736991 751483 121818 275453 592330 130665 585320 443909 088405 781021 728339 > 3166 [i]