Best Known (237−154, 237, s)-Nets in Base 3
(237−154, 237, 58)-Net over F3 — Constructive and digital
Digital (83, 237, 58)-net over F3, using
- net from sequence [i] based on digital (83, 57)-sequence over F3, using
- base reduction for sequences [i] based on digital (13, 57)-sequence over F9, using
- s-reduction based on digital (13, 63)-sequence over F9, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 13 and N(F) ≥ 64, using
- s-reduction based on digital (13, 63)-sequence over F9, using
- base reduction for sequences [i] based on digital (13, 57)-sequence over F9, using
(237−154, 237, 84)-Net over F3 — Digital
Digital (83, 237, 84)-net over F3, using
- t-expansion [i] based on digital (71, 237, 84)-net over F3, using
- net from sequence [i] based on digital (71, 83)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 71 and N(F) ≥ 84, using
- net from sequence [i] based on digital (71, 83)-sequence over F3, using
(237−154, 237, 287)-Net over F3 — Upper bound on s (digital)
There is no digital (83, 237, 288)-net over F3, because
- 1 times m-reduction [i] would yield digital (83, 236, 288)-net over F3, but
- extracting embedded orthogonal array [i] would yield linear OA(3236, 288, F3, 153) (dual of [288, 52, 154]-code), but
- residual code [i] would yield OA(383, 134, S3, 51), but
- the linear programming bound shows that M ≥ 625760 414962 993510 058854 039475 945995 304619 276439 915262 016003 058873 437945 073915 147163 250093 / 135 349071 693745 981183 738057 225151 833299 159482 850560 > 383 [i]
- residual code [i] would yield OA(383, 134, S3, 51), but
- extracting embedded orthogonal array [i] would yield linear OA(3236, 288, F3, 153) (dual of [288, 52, 154]-code), but
(237−154, 237, 362)-Net in Base 3 — Upper bound on s
There is no (83, 237, 363)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 135843 524017 981439 910631 766760 795550 595403 980158 066129 126973 189764 267897 490431 851597 473749 722426 171798 594538 088127 > 3237 [i]