Best Known (62, 62+112, s)-Nets in Base 3
(62, 62+112, 48)-Net over F3 — Constructive and digital
Digital (62, 174, 48)-net over F3, using
- t-expansion [i] based on digital (45, 174, 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
(62, 62+112, 64)-Net over F3 — Digital
Digital (62, 174, 64)-net over F3, using
- t-expansion [i] based on digital (49, 174, 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
(62, 62+112, 247)-Net over F3 — Upper bound on s (digital)
There is no digital (62, 174, 248)-net over F3, because
- 1 times m-reduction [i] would yield digital (62, 173, 248)-net over F3, but
- 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
- extracting embedded orthogonal array [i] would yield linear OA(3173, 248, F3, 111) (dual of [248, 75, 112]-code), but
(62, 62+112, 277)-Net in Base 3 — Upper bound on s
There is no (62, 174, 278)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 109036 616548 753405 042691 228135 852645 420636 925366 127698 793159 218703 865394 304687 463601 > 3174 [i]