Best Known (60, 60+161, s)-Nets in Base 4
(60, 60+161, 66)-Net over F4 — Constructive and digital
Digital (60, 221, 66)-net over F4, using
- t-expansion [i] based on digital (49, 221, 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, 60+161, 91)-Net over F4 — Digital
Digital (60, 221, 91)-net over F4, using
- t-expansion [i] based on digital (50, 221, 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, 60+161, 319)-Net over F4 — Upper bound on s (digital)
There is no digital (60, 221, 320)-net over F4, because
- 1 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, 60+161, 398)-Net in Base 4 — Upper bound on s
There is no (60, 221, 399)-net in base 4, because
- 1 times m-reduction [i] would yield (60, 220, 399)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 2 959592 616138 307097 234881 585150 878241 708776 575004 219938 836080 109130 475607 922769 243013 255784 200582 883301 296938 038641 091320 020273 831624 > 4220 [i]