Best Known (56, 157, s)-Nets in Base 3
(56, 157, 48)-Net over F3 — Constructive and digital
Digital (56, 157, 48)-net over F3, using
- t-expansion [i] based on digital (45, 157, 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
(56, 157, 64)-Net over F3 — Digital
Digital (56, 157, 64)-net over F3, using
- t-expansion [i] based on digital (49, 157, 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
(56, 157, 234)-Net over F3 — Upper bound on s (digital)
There is no digital (56, 157, 235)-net over F3, because
- 2 times m-reduction [i] would yield digital (56, 155, 235)-net over F3, but
- extracting embedded orthogonal array [i] would yield linear OA(3155, 235, F3, 99) (dual of [235, 80, 100]-code), but
- residual code [i] would yield OA(356, 135, S3, 33), but
- the linear programming bound shows that M ≥ 5 540520 515178 119714 998982 762661 978465 988205 121913 455751 263763 784143 824055 345444 731692 711420 873316 188654 321383 010897 917089 561694 445957 308942 731239 278125 / 9558 817773 928640 265890 794601 216127 677021 431399 946162 118946 281454 606414 917120 730244 821409 409342 932190 846015 398469 123832 877827 > 356 [i]
- residual code [i] would yield OA(356, 135, S3, 33), but
- extracting embedded orthogonal array [i] would yield linear OA(3155, 235, F3, 99) (dual of [235, 80, 100]-code), but
(56, 157, 253)-Net in Base 3 — Upper bound on s
There is no (56, 157, 254)-net in base 3, because
- 1 times m-reduction [i] would yield (56, 156, 254)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 282 928608 508732 484101 496231 158623 278375 626868 094672 430128 556311 963370 131549 > 3156 [i]