Best Known (258−188, 258, s)-Nets in Base 4
(258−188, 258, 66)-Net over F4 — Constructive and digital
Digital (70, 258, 66)-net over F4, using
- t-expansion [i] based on digital (49, 258, 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
(258−188, 258, 105)-Net over F4 — Digital
Digital (70, 258, 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
(258−188, 258, 364)-Net over F4 — Upper bound on s (digital)
There is no digital (70, 258, 365)-net over F4, because
- extracting embedded orthogonal array [i] would yield linear OA(4258, 365, F4, 188) (dual of [365, 107, 189]-code), but
- residual code [i] would yield OA(470, 176, S4, 47), but
- 1 times truncation [i] would yield OA(469, 175, S4, 46), but
- the linear programming bound shows that M ≥ 672974 508783 713085 919446 392721 434812 374063 221813 152066 788676 290147 045574 983959 198536 005103 820640 027801 847804 723200 000000 / 1 829712 211075 097010 907956 226624 270704 376891 436886 295247 592787 694007 310519 329949 > 469 [i]
- 1 times truncation [i] would yield OA(469, 175, S4, 46), but
- residual code [i] would yield OA(470, 176, S4, 47), but
(258−188, 258, 461)-Net in Base 4 — Upper bound on s
There is no (70, 258, 462)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 219355 688383 603777 984232 023953 748259 978366 083232 922217 931148 009051 402654 844569 640704 203054 571464 457262 428114 249161 582457 781961 990339 303896 604902 366637 582080 > 4258 [i]