Best Known (173−111, 173, s)-Nets in Base 3
(173−111, 173, 48)-Net over F3 — Constructive and digital
Digital (62, 173, 48)-net over F3, using
- t-expansion [i] based on digital (45, 173, 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
(173−111, 173, 64)-Net over F3 — Digital
Digital (62, 173, 64)-net over F3, using
- t-expansion [i] based on digital (49, 173, 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
(173−111, 173, 247)-Net over F3 — Upper bound on s (digital)
There is no digital (62, 173, 248)-net over F3, because
- extracting embedded orthogonal array [i] would yield linear OA(3173, 248, F3, 111) (dual of [248, 75, 112]-code), but
- residual code [i] would yield OA(362, 136, S3, 37), but
- the linear programming bound shows that M ≥ 25 107906 551961 934560 769174 614244 000286 915780 821438 335401 839731 269118 018422 936529 743703 225263 610816 590271 097423 222819 087158 969802 653064 295993 622444 421992 834724 094091 277181 403391 938201 079579 777340 758040 595510 894725 941611 026581 156837 394107 895869 075792 152530 290305 / 60 978394 122954 076492 784397 168062 353216 564275 839619 092129 650145 178598 586317 693508 759218 909634 383305 197348 832874 724604 670145 610063 425705 187278 056568 354935 117924 137745 719797 069021 950479 890815 422812 433736 033606 282594 894113 361942 802593 > 362 [i]
- residual code [i] would yield OA(362, 136, S3, 37), but
(173−111, 173, 280)-Net in Base 3 — Upper bound on s
There is no (62, 173, 281)-net in base 3, because
- 1 times m-reduction [i] would yield (62, 172, 281)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 13210 909466 313498 746872 324013 469959 971810 474377 700924 510804 730668 728708 655478 680875 > 3172 [i]