Best Known (60, 223, s)-Nets in Base 4
(60, 223, 66)-Net over F4 — Constructive and digital
Digital (60, 223, 66)-net over F4, using
- t-expansion [i] based on digital (49, 223, 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
(60, 223, 91)-Net over F4 — Digital
Digital (60, 223, 91)-net over F4, using
- t-expansion [i] based on digital (50, 223, 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
(60, 223, 319)-Net over F4 — Upper bound on s (digital)
There is no digital (60, 223, 320)-net over F4, because
- 3 times m-reduction [i] would yield digital (60, 220, 320)-net over F4, but
- extracting embedded orthogonal array [i] would yield linear OA(4220, 320, F4, 160) (dual of [320, 100, 161]-code), but
- residual code [i] would yield OA(460, 159, S4, 40), but
- the linear programming bound shows that M ≥ 89256 467878 907972 172600 180176 154060 802582 684769 875981 969312 131469 276806 223240 396340 011634 339610 624000 / 64975 225388 099566 116025 141010 137689 743517 140250 845468 320822 687201 > 460 [i]
- residual code [i] would yield OA(460, 159, S4, 40), but
- extracting embedded orthogonal array [i] would yield linear OA(4220, 320, F4, 160) (dual of [320, 100, 161]-code), but
(60, 223, 397)-Net in Base 4 — Upper bound on s
There is no (60, 223, 398)-net in base 4, because
- 1 times m-reduction [i] would yield (60, 222, 398)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 48 441955 410244 709345 927764 305532 577396 863121 322174 938601 925678 145937 031450 047455 552457 208681 616020 469283 489947 991752 083942 325496 851800 > 4222 [i]