Best Known (51, 148, s)-Nets in Base 3
(51, 148, 48)-Net over F3 — Constructive and digital
Digital (51, 148, 48)-net over F3, using
- t-expansion [i] based on digital (45, 148, 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
(51, 148, 64)-Net over F3 — Digital
Digital (51, 148, 64)-net over F3, using
- t-expansion [i] based on digital (49, 148, 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
(51, 148, 187)-Net over F3 — Upper bound on s (digital)
There is no digital (51, 148, 188)-net over F3, because
- 1 times m-reduction [i] would yield digital (51, 147, 188)-net over F3, but
- extracting embedded orthogonal array [i] would yield linear OA(3147, 188, F3, 96) (dual of [188, 41, 97]-code), but
- residual code [i] would yield OA(351, 91, S3, 32), but
- the linear programming bound shows that M ≥ 3 796003 121606 659684 644081 485270 254802 809922 702083 145243 272087 278428 845740 570091 586803 340207 313946 354006 591960 625547 / 1 756668 470243 430645 991636 297238 233510 101788 678484 940922 765709 380449 381903 813470 038571 507115 > 351 [i]
- residual code [i] would yield OA(351, 91, S3, 32), but
- extracting embedded orthogonal array [i] would yield linear OA(3147, 188, F3, 96) (dual of [188, 41, 97]-code), but
(51, 148, 226)-Net in Base 3 — Upper bound on s
There is no (51, 148, 227)-net in base 3, because
- 1 times m-reduction [i] would yield (51, 147, 227)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 14568 955706 033181 027131 505736 047702 326991 066991 924636 525984 351902 768801 > 3147 [i]