Best Known (81, 81+154, s)-Nets in Base 3
(81, 81+154, 56)-Net over F3 — Constructive and digital
Digital (81, 235, 56)-net over F3, using
- net from sequence [i] based on digital (81, 55)-sequence over F3, using
- base reduction for sequences [i] based on digital (13, 55)-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, 55)-sequence over F9, using
(81, 81+154, 84)-Net over F3 — Digital
Digital (81, 235, 84)-net over F3, using
- t-expansion [i] based on digital (71, 235, 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
(81, 81+154, 266)-Net over F3 — Upper bound on s (digital)
There is no digital (81, 235, 267)-net over F3, because
- 1 times m-reduction [i] would yield digital (81, 234, 267)-net over F3, but
- extracting embedded orthogonal array [i] would yield linear OA(3234, 267, F3, 153) (dual of [267, 33, 154]-code), but
- residual code [i] would yield OA(381, 113, S3, 51), but
- the linear programming bound shows that M ≥ 23713 080736 201971 210054 022630 767751 231756 211462 816760 345549 / 44 946009 615547 625000 > 381 [i]
- residual code [i] would yield OA(381, 113, S3, 51), but
- extracting embedded orthogonal array [i] would yield linear OA(3234, 267, F3, 153) (dual of [267, 33, 154]-code), but
(81, 81+154, 350)-Net in Base 3 — Upper bound on s
There is no (81, 235, 351)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 15330 027230 855250 222328 336065 830407 571587 593219 181973 461457 115065 909888 374538 261772 424192 379872 620730 380867 966759 > 3235 [i]