Best Known (74−69, 74, s)-Nets in Base 16
(74−69, 74, 49)-Net over F16 — Constructive and digital
Digital (5, 74, 49)-net over F16, using
- net from sequence [i] based on digital (5, 48)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 5 and N(F) ≥ 49, using
(74−69, 74, 158)-Net over F16 — Upper bound on s (digital)
There is no digital (5, 74, 159)-net over F16, because
- 5 times m-reduction [i] would yield digital (5, 69, 159)-net over F16, but
- extracting embedded orthogonal array [i] would yield linear OA(1669, 159, F16, 64) (dual of [159, 90, 65]-code), but
- residual code [i] would yield OA(165, 94, S16, 4), but
- the linear programming bound shows that M ≥ 2179 760128 / 2071 > 165 [i]
- residual code [i] would yield OA(165, 94, S16, 4), but
- extracting embedded orthogonal array [i] would yield linear OA(1669, 159, F16, 64) (dual of [159, 90, 65]-code), but
(74−69, 74, 163)-Net in Base 16 — Upper bound on s
There is no (5, 74, 164)-net in base 16, because
- extracting embedded orthogonal array [i] would yield OA(1674, 164, S16, 69), but
- the linear programming bound shows that M ≥ 16135 809135 562727 076026 699590 876724 126954 604517 745820 850758 529681 635101 286350 957220 660860 477779 285138 697953 283553 310810 880043 528897 953792 / 123530 991953 579422 726455 608763 912141 416019 177557 > 1674 [i]