Best Known (216−158, 216, s)-Nets in Base 4
(216−158, 216, 66)-Net over F4 — Constructive and digital
Digital (58, 216, 66)-net over F4, using
- t-expansion [i] based on digital (49, 216, 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
(216−158, 216, 91)-Net over F4 — Digital
Digital (58, 216, 91)-net over F4, using
- t-expansion [i] based on digital (50, 216, 91)-net over F4, using
- net from sequence [i] based on digital (50, 90)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 50 and N(F) ≥ 91, using
- net from sequence [i] based on digital (50, 90)-sequence over F4, using
(216−158, 216, 305)-Net over F4 — Upper bound on s (digital)
There is no digital (58, 216, 306)-net over F4, because
- 2 times m-reduction [i] would yield digital (58, 214, 306)-net over F4, but
- extracting embedded orthogonal array [i] would yield linear OA(4214, 306, F4, 156) (dual of [306, 92, 157]-code), but
- residual code [i] would yield OA(458, 149, S4, 39), but
- the linear programming bound shows that M ≥ 687 429911 202260 642762 810713 800269 676548 725551 961062 930193 746529 711584 285763 934617 600000 / 7739 058251 757365 326970 583055 442734 658951 141232 199899 > 458 [i]
- residual code [i] would yield OA(458, 149, S4, 39), but
- extracting embedded orthogonal array [i] would yield linear OA(4214, 306, F4, 156) (dual of [306, 92, 157]-code), but
(216−158, 216, 384)-Net in Base 4 — Upper bound on s
There is no (58, 216, 385)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 12913 455255 718527 225082 270219 195541 474219 871641 741173 393578 647504 358589 844083 822328 753984 850952 934489 591964 670158 108029 037977 505664 > 4216 [i]