Best Known (75−72, 75, s)-Nets in Base 25
(75−72, 75, 52)-Net over F25 — Constructive and digital
Digital (3, 75, 52)-net over F25, using
- net from sequence [i] based on digital (3, 51)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 3 and N(F) ≥ 52, using
(75−72, 75, 56)-Net over F25 — Digital
Digital (3, 75, 56)-net over F25, using
- net from sequence [i] based on digital (3, 55)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 3 and N(F) ≥ 56, using
(75−72, 75, 147)-Net over F25 — Upper bound on s (digital)
There is no digital (3, 75, 148)-net over F25, because
- 5 times m-reduction [i] would yield digital (3, 70, 148)-net over F25, but
- extracting embedded orthogonal array [i] would yield linear OA(2570, 148, F25, 67) (dual of [148, 78, 68]-code), but
- construction Y1 [i] would yield
- OA(2569, 73, S25, 67), but
- the linear programming bound shows that M ≥ 24178 564222 846123 668546 399721 917528 895562 151649 914153 732205 158997 548011 257094 913162 291049 957275 390625 / 7242 > 2569 [i]
- linear OA(2578, 148, F25, 75) (dual of [148, 70, 76]-code), but
- discarding factors / shortening the dual code would yield linear OA(2578, 103, F25, 75) (dual of [103, 25, 76]-code), but
- residual code [i] would yield OA(253, 27, S25, 3), but
- discarding factors / shortening the dual code would yield linear OA(2578, 103, F25, 75) (dual of [103, 25, 76]-code), but
- OA(2569, 73, S25, 67), but
- construction Y1 [i] would yield
- extracting embedded orthogonal array [i] would yield linear OA(2570, 148, F25, 67) (dual of [148, 78, 68]-code), but
(75−72, 75, 186)-Net in Base 25 — Upper bound on s
There is no (3, 75, 187)-net in base 25, because
- extracting embedded orthogonal array [i] would yield OA(2575, 187, S25, 72), but
- the linear programming bound shows that M ≥ 360 053330 434353 022072 023224 409010 593389 546484 777789 073714 738151 034409 199911 260146 008732 362099 546975 287012 173794 209957 122802 734375 / 503659 960533 818113 126807 > 2575 [i]