Best Known (169−126, 169, s)-Nets in Base 4
(169−126, 169, 56)-Net over F4 — Constructive and digital
Digital (43, 169, 56)-net over F4, using
- t-expansion [i] based on digital (33, 169, 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
(169−126, 169, 75)-Net over F4 — Digital
Digital (43, 169, 75)-net over F4, using
- t-expansion [i] based on digital (40, 169, 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
(169−126, 169, 208)-Net over F4 — Upper bound on s (digital)
There is no digital (43, 169, 209)-net over F4, because
- 2 times m-reduction [i] would yield digital (43, 167, 209)-net over F4, but
- extracting embedded orthogonal array [i] would yield linear OA(4167, 209, F4, 124) (dual of [209, 42, 125]-code), but
- residual code [i] would yield OA(443, 84, S4, 31), but
- the linear programming bound shows that M ≥ 743 432824 923585 138466 079893 608385 692163 936496 473876 252564 706270 846821 427655 153621 009913 103585 162421 132960 451704 728431 099904 / 9 052373 288486 345160 206941 767894 606217 335850 613093 145925 916389 354432 331859 732314 014078 839167 611385 > 443 [i]
- residual code [i] would yield OA(443, 84, S4, 31), but
- extracting embedded orthogonal array [i] would yield linear OA(4167, 209, F4, 124) (dual of [209, 42, 125]-code), but
(169−126, 169, 284)-Net in Base 4 — Upper bound on s
There is no (43, 169, 285)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 587040 805400 335556 987231 893937 402070 047937 693765 900007 374319 738329 970889 222988 451753 395642 700804 660480 > 4169 [i]