Best Known (158−104, 158, s)-Nets in Base 3
(158−104, 158, 48)-Net over F3 — Constructive and digital
Digital (54, 158, 48)-net over F3, using
- t-expansion [i] based on digital (45, 158, 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
(158−104, 158, 64)-Net over F3 — Digital
Digital (54, 158, 64)-net over F3, using
- t-expansion [i] based on digital (49, 158, 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
(158−104, 158, 193)-Net over F3 — Upper bound on s (digital)
There is no digital (54, 158, 194)-net over F3, because
- 2 times m-reduction [i] would yield digital (54, 156, 194)-net over F3, but
- extracting embedded orthogonal array [i] would yield linear OA(3156, 194, F3, 102) (dual of [194, 38, 103]-code), but
- residual code [i] would yield OA(354, 91, S3, 34), but
- the linear programming bound shows that M ≥ 474 977782 539634 784051 796341 060967 928894 025210 202817 107183 513784 170892 978866 150753 / 7 995136 378245 997436 849186 466159 148790 740510 898632 999345 > 354 [i]
- residual code [i] would yield OA(354, 91, S3, 34), but
- extracting embedded orthogonal array [i] would yield linear OA(3156, 194, F3, 102) (dual of [194, 38, 103]-code), but
(158−104, 158, 236)-Net in Base 3 — Upper bound on s
There is no (54, 158, 237)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 2488 162787 225524 997990 215223 941656 684279 003637 311281 346054 364579 861081 994961 > 3158 [i]