Best Known (153−98, 153, s)-Nets in Base 3
(153−98, 153, 48)-Net over F3 — Constructive and digital
Digital (55, 153, 48)-net over F3, using
- t-expansion [i] based on digital (45, 153, 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
(153−98, 153, 64)-Net over F3 — Digital
Digital (55, 153, 64)-net over F3, using
- t-expansion [i] based on digital (49, 153, 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
(153−98, 153, 232)-Net over F3 — Upper bound on s (digital)
There is no digital (55, 153, 233)-net over F3, because
- 2 times m-reduction [i] would yield digital (55, 151, 233)-net over F3, but
- extracting embedded orthogonal array [i] would yield linear OA(3151, 233, F3, 96) (dual of [233, 82, 97]-code), but
- residual code [i] would yield OA(355, 136, S3, 32), but
- the linear programming bound shows that M ≥ 3692 381881 435086 561083 538512 781551 192892 214805 966690 241846 076111 760291 378162 952668 981065 231048 784442 555038 600643 905023 157760 / 19 467037 909253 478265 941280 359023 466094 270607 906474 140765 003376 713759 364882 992681 224990 199775 364421 > 355 [i]
- residual code [i] would yield OA(355, 136, S3, 32), but
- extracting embedded orthogonal array [i] would yield linear OA(3151, 233, F3, 96) (dual of [233, 82, 97]-code), but
(153−98, 153, 249)-Net in Base 3 — Upper bound on s
There is no (55, 153, 250)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 10 472571 986212 094490 567229 716819 754614 263581 059276 221319 205729 521703 584021 > 3153 [i]