Best Known (242−156, 242, s)-Nets in Base 3
(242−156, 242, 61)-Net over F3 — Constructive and digital
Digital (86, 242, 61)-net over F3, using
- net from sequence [i] based on digital (86, 60)-sequence over F3, using
- base reduction for sequences [i] based on digital (13, 60)-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, 60)-sequence over F9, using
(242−156, 242, 84)-Net over F3 — Digital
Digital (86, 242, 84)-net over F3, using
- t-expansion [i] based on digital (71, 242, 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
(242−156, 242, 306)-Net over F3 — Upper bound on s (digital)
There is no digital (86, 242, 307)-net over F3, because
- extracting embedded orthogonal array [i] would yield linear OA(3242, 307, F3, 156) (dual of [307, 65, 157]-code), but
- residual code [i] would yield OA(386, 150, S3, 52), but
- the linear programming bound shows that M ≥ 45 375615 652569 081636 015380 745443 018934 304283 021352 151426 389016 725239 072381 376675 411874 549461 398308 950056 387797 392972 439185 682172 765235 884412 994552 176933 079882 633952 671620 169056 236224 985173 977239 215077 245807 038742 365477 871088 318198 915432 / 411 139499 203801 445198 183759 841057 049789 374017 649665 468263 260057 814101 412003 776038 812811 220305 805521 531916 470583 053309 910406 226958 416674 671701 921653 986187 454828 049325 832567 286120 869274 575255 729375 > 386 [i]
- residual code [i] would yield OA(386, 150, S3, 52), but
(242−156, 242, 378)-Net in Base 3 — Upper bound on s
There is no (86, 242, 379)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 30 500906 704292 767982 848345 412700 286621 507031 919531 205297 287107 342255 321943 660227 974194 114000 759487 880421 300315 833085 > 3242 [i]