Best Known (31, 31+42, s)-Nets in Base 3
(31, 31+42, 37)-Net over F3 — Constructive and digital
Digital (31, 73, 37)-net over F3, using
- t-expansion [i] based on digital (27, 73, 37)-net over F3, using
- net from sequence [i] based on digital (27, 36)-sequence over F3, using
- Niederreiter–Xing sequence construction III based on the algebraic function field F/F3 with g(F) = 26, N(F) = 36, and 1 place with degree 2 [i] based on function field F/F3 with g(F) = 26 and N(F) ≥ 36, using an explicitly constructive algebraic function field [i]
- net from sequence [i] based on digital (27, 36)-sequence over F3, using
(31, 31+42, 42)-Net over F3 — Digital
Digital (31, 73, 42)-net over F3, using
- t-expansion [i] based on digital (29, 73, 42)-net over F3, using
- net from sequence [i] based on digital (29, 41)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 29 and N(F) ≥ 42, using
- net from sequence [i] based on digital (29, 41)-sequence over F3, using
(31, 31+42, 178)-Net in Base 3 — Upper bound on s
There is no (31, 73, 179)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 74757 258256 722866 753863 989610 832319 > 373 [i]
- extracting embedded orthogonal array [i] would yield OA(373, 179, S3, 42), but
- the linear programming bound shows that M ≥ 1655 853606 621594 847365 162793 421685 754880 172836 255628 739509 006491 538956 023548 309718 301066 295705 004185 498409 491414 287212 260645 557088 062619 662927 494772 522986 898423 456227 215054 895585 942341 350400 / 23155 044107 809386 801034 895150 368402 409396 608253 215274 583956 452423 594904 331538 297952 444628 898814 412153 928733 894090 289314 856067 063972 880940 913479 439182 047891 > 373 [i]