Best Known (58, 58+103, s)-Nets in Base 3
(58, 58+103, 48)-Net over F3 — Constructive and digital
Digital (58, 161, 48)-net over F3, using
- t-expansion [i] based on digital (45, 161, 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
(58, 58+103, 64)-Net over F3 — Digital
Digital (58, 161, 64)-net over F3, using
- t-expansion [i] based on digital (49, 161, 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
(58, 58+103, 240)-Net over F3 — Upper bound on s (digital)
There is no digital (58, 161, 241)-net over F3, because
- 1 times m-reduction [i] would yield digital (58, 160, 241)-net over F3, but
- extracting embedded orthogonal array [i] would yield linear OA(3160, 241, F3, 102) (dual of [241, 81, 103]-code), but
- residual code [i] would yield OA(358, 138, S3, 34), but
- the linear programming bound shows that M ≥ 127476 152647 109858 836964 999510 941941 258810 174509 185909 510810 236158 485486 586777 642310 167217 093997 490577 412163 979598 552712 810081 135105 109375 / 26 664339 762561 735625 225371 208573 781952 042183 497535 838408 714653 075655 100686 712415 616991 631039 991283 863425 318551 > 358 [i]
- residual code [i] would yield OA(358, 138, S3, 34), but
- extracting embedded orthogonal array [i] would yield linear OA(3160, 241, F3, 102) (dual of [241, 81, 103]-code), but
(58, 58+103, 264)-Net in Base 3 — Upper bound on s
There is no (58, 161, 265)-net in base 3, because
- 1 times m-reduction [i] would yield (58, 160, 265)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 24543 550232 285144 562927 346616 404170 275920 259504 834221 458823 834969 713533 330619 > 3160 [i]