Best Known (81−78, 81, s)-Nets in Base 25
(81−78, 81, 52)-Net over F25 — Constructive and digital
Digital (3, 81, 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
(81−78, 81, 56)-Net over F25 — Digital
Digital (3, 81, 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
(81−78, 81, 102)-Net over F25 — Upper bound on s (digital)
There is no digital (3, 81, 103)-net over F25, because
- 3 times m-reduction [i] would yield digital (3, 78, 103)-net over F25, but
- extracting embedded orthogonal array [i] 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
- extracting embedded orthogonal array [i] would yield linear OA(2578, 103, F25, 75) (dual of [103, 25, 76]-code), but
(81−78, 81, 170)-Net in Base 25 — Upper bound on s
There is no (3, 81, 171)-net in base 25, because
- extracting embedded orthogonal array [i] would yield OA(2581, 171, S25, 78), but
- the linear programming bound shows that M ≥ 23205 784249 597023 772443 087743 632742 305389 821673 152514 637717 572494 013805 538409 460327 522440 800163 995577 864994 857009 151019 155979 156494 140625 / 133516 675673 119357 572553 > 2581 [i]