Best Known (237−158, 237, s)-Nets in Base 3
(237−158, 237, 54)-Net over F3 — Constructive and digital
Digital (79, 237, 54)-net over F3, using
- net from sequence [i] based on digital (79, 53)-sequence over F3, using
- base reduction for sequences [i] based on digital (13, 53)-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, 53)-sequence over F9, using
(237−158, 237, 84)-Net over F3 — Digital
Digital (79, 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
(237−158, 237, 250)-Net over F3 — Upper bound on s (digital)
There is no digital (79, 237, 251)-net over F3, because
- 2 times m-reduction [i] would yield digital (79, 235, 251)-net over F3, but
- extracting embedded orthogonal array [i] would yield linear OA(3235, 251, F3, 156) (dual of [251, 16, 157]-code), but
- residual code [i] would yield OA(379, 94, S3, 52), but
- 1 times truncation [i] would yield OA(378, 93, S3, 51), but
- the linear programming bound shows that M ≥ 2117 985825 855951 548682 979121 686352 501183 391624 / 117 447583 > 378 [i]
- 1 times truncation [i] would yield OA(378, 93, S3, 51), but
- residual code [i] would yield OA(379, 94, S3, 52), but
- extracting embedded orthogonal array [i] would yield linear OA(3235, 251, F3, 156) (dual of [251, 16, 157]-code), but
(237−158, 237, 335)-Net in Base 3 — Upper bound on s
There is no (79, 237, 336)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 138011 006303 335228 766492 843975 611808 447575 001049 078787 382329 397035 033721 239601 777383 613655 646842 834011 308000 232385 > 3237 [i]