Best Known (224−146, 224, s)-Nets in Base 3
(224−146, 224, 53)-Net over F3 — Constructive and digital
Digital (78, 224, 53)-net over F3, using
- net from sequence [i] based on digital (78, 52)-sequence over F3, using
- base reduction for sequences [i] based on digital (13, 52)-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, 52)-sequence over F9, using
(224−146, 224, 84)-Net over F3 — Digital
Digital (78, 224, 84)-net over F3, using
- t-expansion [i] based on digital (71, 224, 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
(224−146, 224, 272)-Net over F3 — Upper bound on s (digital)
There is no digital (78, 224, 273)-net over F3, because
- 2 times m-reduction [i] would yield digital (78, 222, 273)-net over F3, but
- extracting embedded orthogonal array [i] would yield linear OA(3222, 273, F3, 144) (dual of [273, 51, 145]-code), but
- residual code [i] would yield OA(378, 128, S3, 48), but
- the linear programming bound shows that M ≥ 6 575490 843517 549614 037804 755790 886921 388297 503372 467014 072409 123799 884455 544386 348644 065807 588730 807152 499714 277823 942029 / 372064 864657 803488 591486 645857 495615 261692 996975 549874 881531 815830 072025 751212 712500 > 378 [i]
- residual code [i] would yield OA(378, 128, S3, 48), but
- extracting embedded orthogonal array [i] would yield linear OA(3222, 273, F3, 144) (dual of [273, 51, 145]-code), but
(224−146, 224, 340)-Net in Base 3 — Upper bound on s
There is no (78, 224, 341)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 89206 169642 822043 196667 713455 617774 058708 905606 378661 174805 759362 002449 141497 729141 225860 106608 935443 076107 > 3224 [i]