Best Known (159−118, 159, s)-Nets in Base 4
(159−118, 159, 56)-Net over F4 — Constructive and digital
Digital (41, 159, 56)-net over F4, using
- t-expansion [i] based on digital (33, 159, 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
(159−118, 159, 75)-Net over F4 — Digital
Digital (41, 159, 75)-net over F4, using
- t-expansion [i] based on digital (40, 159, 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
(159−118, 159, 208)-Net over F4 — Upper bound on s (digital)
There is no digital (41, 159, 209)-net over F4, because
- 2 times m-reduction [i] would yield digital (41, 157, 209)-net over F4, but
- extracting embedded orthogonal array [i] would yield linear OA(4157, 209, F4, 116) (dual of [209, 52, 117]-code), but
- residual code [i] would yield OA(441, 92, S4, 29), but
- the linear programming bound shows that M ≥ 8291 138749 676505 103841 959950 541753 992228 411787 128227 036594 724129 251790 946304 / 1649 446049 086149 969896 114658 918272 865279 146667 501047 > 441 [i]
- residual code [i] would yield OA(441, 92, S4, 29), but
- extracting embedded orthogonal array [i] would yield linear OA(4157, 209, F4, 116) (dual of [209, 52, 117]-code), but
(159−118, 159, 272)-Net in Base 4 — Upper bound on s
There is no (41, 159, 273)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 542788 104776 016167 487161 968872 148714 513851 125132 060833 580944 245976 991177 893347 486816 309706 007296 > 4159 [i]