Best Known (218−158, 218, s)-Nets in Base 4
(218−158, 218, 66)-Net over F4 — Constructive and digital
Digital (60, 218, 66)-net over F4, using
- t-expansion [i] based on digital (49, 218, 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
(218−158, 218, 91)-Net over F4 — Digital
Digital (60, 218, 91)-net over F4, using
- t-expansion [i] based on digital (50, 218, 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
(218−158, 218, 336)-Net over F4 — Upper bound on s (digital)
There is no digital (60, 218, 337)-net over F4, because
- 2 times m-reduction [i] would yield digital (60, 216, 337)-net over F4, but
- extracting embedded orthogonal array [i] would yield linear OA(4216, 337, F4, 156) (dual of [337, 121, 157]-code), but
- residual code [i] would yield OA(460, 180, S4, 39), but
- the linear programming bound shows that M ≥ 64 592000 643692 089079 054839 691926 765927 402602 779909 340826 367902 235627 537241 992340 655885 516800 / 46 756581 550576 177840 289976 743881 661381 961798 582533 083517 > 460 [i]
- residual code [i] would yield OA(460, 180, S4, 39), but
- extracting embedded orthogonal array [i] would yield linear OA(4216, 337, F4, 156) (dual of [337, 121, 157]-code), but
(218−158, 218, 399)-Net in Base 4 — Upper bound on s
There is no (60, 218, 400)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 178356 897936 227076 697561 198892 017558 508356 020903 759080 275605 042719 473450 553066 017264 672788 954806 984489 544798 480338 340348 876437 989294 > 4218 [i]