Best Known (42, 42+120, s)-Nets in Base 4
(42, 42+120, 56)-Net over F4 — Constructive and digital
Digital (42, 162, 56)-net over F4, using
- t-expansion [i] based on digital (33, 162, 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
(42, 42+120, 75)-Net over F4 — Digital
Digital (42, 162, 75)-net over F4, using
- t-expansion [i] based on digital (40, 162, 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
- net from sequence [i] based on digital (40, 74)-sequence over F4, using
(42, 42+120, 208)-Net over F4 — Upper bound on s (digital)
There is no digital (42, 162, 209)-net over F4, because
- extracting embedded orthogonal array [i] would yield linear OA(4162, 209, F4, 120) (dual of [209, 47, 121]-code), but
- residual code [i] would yield OA(442, 88, S4, 30), but
- the linear programming bound shows that M ≥ 89 605050 492137 134852 017163 586838 439030 395196 856884 186345 777644 655144 707653 985466 546419 715121 697993 523200 / 4 627392 744873 205288 863270 778823 516250 097275 954094 267550 824480 805099 989747 440773 > 442 [i]
- residual code [i] would yield OA(442, 88, S4, 30), but
(42, 42+120, 279)-Net in Base 4 — Upper bound on s
There is no (42, 162, 280)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 36 929237 982064 952632 215614 280607 954491 470505 459897 962608 619646 429661 394003 169736 110460 310177 484640 > 4162 [i]