Best Known (40, 153, s)-Nets in Base 4
(40, 153, 56)-Net over F4 — Constructive and digital
Digital (40, 153, 56)-net over F4, using
- t-expansion [i] based on digital (33, 153, 56)-net over F4, using
- net from sequence [i] based on digital (33, 55)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 33 and N(F) ≥ 56, using
- F5 from the 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) = 33 and N(F) ≥ 56, using
- net from sequence [i] based on digital (33, 55)-sequence over F4, using
(40, 153, 75)-Net over F4 — Digital
Digital (40, 153, 75)-net over F4, using
- net from sequence [i] based on digital (40, 74)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 40 and N(F) ≥ 75, using
(40, 153, 208)-Net over F4 — Upper bound on s (digital)
There is no digital (40, 153, 209)-net over F4, because
- 1 times m-reduction [i] would yield digital (40, 152, 209)-net over F4, but
- extracting embedded orthogonal array [i] would yield linear OA(4152, 209, F4, 112) (dual of [209, 57, 113]-code), but
- residual code [i] would yield OA(440, 96, S4, 28), but
- the linear programming bound shows that M ≥ 3 041221 052768 594873 805769 980859 249005 438281 138205 879215 063040 / 2 375396 786313 270370 198219 858144 081283 > 440 [i]
- residual code [i] would yield OA(440, 96, S4, 28), but
- extracting embedded orthogonal array [i] would yield linear OA(4152, 209, F4, 112) (dual of [209, 57, 113]-code), but
(40, 153, 267)-Net in Base 4 — Upper bound on s
There is no (40, 153, 268)-net in base 4, because
- 1 times m-reduction [i] would yield (40, 152, 268)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 33 045312 937429 651775 445853 143568 210132 981226 863868 840522 596667 667220 459925 808687 136396 127920 > 4152 [i]