Best Known (59, 217, s)-Nets in Base 4
(59, 217, 66)-Net over F4 — Constructive and digital
Digital (59, 217, 66)-net over F4, using
- t-expansion [i] based on digital (49, 217, 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
(59, 217, 91)-Net over F4 — Digital
Digital (59, 217, 91)-net over F4, using
- t-expansion [i] based on digital (50, 217, 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
(59, 217, 321)-Net over F4 — Upper bound on s (digital)
There is no digital (59, 217, 322)-net over F4, because
- 2 times m-reduction [i] would yield digital (59, 215, 322)-net over F4, but
- extracting embedded orthogonal array [i] would yield linear OA(4215, 322, F4, 156) (dual of [322, 107, 157]-code), but
- residual code [i] would yield OA(459, 165, S4, 39), but
- the linear programming bound shows that M ≥ 3078 489369 750206 722287 954878 459506 651865 539185 921556 985227 028939 036190 618883 341680 640000 / 8667 454957 234129 743329 832681 955517 588182 622985 872027 > 459 [i]
- residual code [i] would yield OA(459, 165, S4, 39), but
- extracting embedded orthogonal array [i] would yield linear OA(4215, 322, F4, 156) (dual of [322, 107, 157]-code), but
(59, 217, 391)-Net in Base 4 — Upper bound on s
There is no (59, 217, 392)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 44456 803645 445179 172497 217203 575409 119714 799757 695842 937524 400370 045131 519413 482040 157096 588911 123790 955652 003071 506367 746553 752816 > 4217 [i]