Best Known (71−68, 71, s)-Nets in Base 25
(71−68, 71, 52)-Net over F25 — Constructive and digital
Digital (3, 71, 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
(71−68, 71, 56)-Net over F25 — Digital
Digital (3, 71, 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
(71−68, 71, 147)-Net over F25 — Upper bound on s (digital)
There is no digital (3, 71, 148)-net over F25, because
- 1 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
(71−68, 71, 225)-Net in Base 25 — Upper bound on s
There is no (3, 71, 226)-net in base 25, because
- extracting embedded orthogonal array [i] would yield OA(2571, 226, S25, 68), but
- the linear programming bound shows that M ≥ 35 332779 205869 297739 243813 593843 116483 595828 585084 351776 469108 348080 397891 976026 612803 618576 208464 219234 883785 247802 734375 / 19614 710635 502434 371223 > 2571 [i]