Best Known (247−180, 247, s)-Nets in Base 4
(247−180, 247, 66)-Net over F4 — Constructive and digital
Digital (67, 247, 66)-net over F4, using
- t-expansion [i] based on digital (49, 247, 66)-net over F4, using
- net from sequence [i] based on digital (49, 65)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 49 and N(F) ≥ 66, using
- T6 from the second tower of function fields by GarcÃa and Stichtenoth over F4 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 49 and N(F) ≥ 66, using
- net from sequence [i] based on digital (49, 65)-sequence over F4, using
(247−180, 247, 99)-Net over F4 — Digital
Digital (67, 247, 99)-net over F4, using
- t-expansion [i] based on digital (61, 247, 99)-net over F4, using
- net from sequence [i] based on digital (61, 98)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 61 and N(F) ≥ 99, using
- net from sequence [i] based on digital (61, 98)-sequence over F4, using
(247−180, 247, 345)-Net over F4 — Upper bound on s (digital)
There is no digital (67, 247, 346)-net over F4, because
- extracting embedded orthogonal array [i] would yield linear OA(4247, 346, F4, 180) (dual of [346, 99, 181]-code), but
- residual code [i] would yield OA(467, 165, S4, 45), but
- the linear programming bound shows that M ≥ 20699 363194 875685 646876 418367 204535 739040 572858 578915 564075 720939 347504 204480 515182 643491 673405 319238 146623 194291 154149 560162 674592 972800 / 923774 886925 816898 766657 500544 462579 832080 182768 042463 529757 381333 868390 078991 844533 113172 156937 > 467 [i]
- residual code [i] would yield OA(467, 165, S4, 45), but
(247−180, 247, 442)-Net in Base 4 — Upper bound on s
There is no (67, 247, 443)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 53749 243544 968330 239309 492661 910147 479030 258862 058636 672472 160606 593537 842478 856044 476588 898873 807344 095875 980654 772702 563000 907205 669968 150651 009210 > 4247 [i]