Best Known (72, 72+182, s)-Nets in Base 4
(72, 72+182, 66)-Net over F4 — Constructive and digital
Digital (72, 254, 66)-net over F4, using
- t-expansion [i] based on digital (49, 254, 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
(72, 72+182, 105)-Net over F4 — Digital
Digital (72, 254, 105)-net over F4, using
- t-expansion [i] based on digital (70, 254, 105)-net over F4, using
- net from sequence [i] based on digital (70, 104)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 70 and N(F) ≥ 105, using
- net from sequence [i] based on digital (70, 104)-sequence over F4, using
(72, 72+182, 430)-Net over F4 — Upper bound on s (digital)
There is no digital (72, 254, 431)-net over F4, because
- 2 times m-reduction [i] would yield digital (72, 252, 431)-net over F4, but
- extracting embedded orthogonal array [i] would yield linear OA(4252, 431, F4, 180) (dual of [431, 179, 181]-code), but
- residual code [i] would yield OA(472, 250, S4, 45), but
- the linear programming bound shows that M ≥ 844 253129 128337 968002 738138 104958 795392 851484 110411 486022 086287 533951 460726 969110 480212 292670 586880 / 36 177347 187970 342236 372937 633185 534214 836452 245739 305493 > 472 [i]
- residual code [i] would yield OA(472, 250, S4, 45), but
- extracting embedded orthogonal array [i] would yield linear OA(4252, 431, F4, 180) (dual of [431, 179, 181]-code), but
(72, 72+182, 481)-Net in Base 4 — Upper bound on s
There is no (72, 254, 482)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 839 123640 493358 490432 604711 464308 408313 531657 231526 082673 340664 540223 561763 536001 564391 044758 380718 395245 767051 082106 644490 930761 969444 550461 750529 293456 > 4254 [i]