Best Known (54−49, 54, s)-Nets in Base 5
(54−49, 54, 20)-Net over F5 — Constructive and digital
Digital (5, 54, 20)-net over F5, using
- net from sequence [i] based on digital (5, 19)-sequence over F5, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 5 and N(F) ≥ 20, using
(54−49, 54, 32)-Net over F5 — Upper bound on s (digital)
There is no digital (5, 54, 33)-net over F5, because
- 25 times m-reduction [i] would yield digital (5, 29, 33)-net over F5, but
- extracting embedded orthogonal array [i] would yield linear OA(529, 33, F5, 24) (dual of [33, 4, 25]-code), but
(54−49, 54, 34)-Net in Base 5 — Upper bound on s
There is no (5, 54, 35)-net in base 5, because
- 24 times m-reduction [i] would yield (5, 30, 35)-net in base 5, but
- extracting embedded orthogonal array [i] would yield OA(530, 35, S5, 25), but
- the linear programming bound shows that M ≥ 395812 094211 578369 140625 / 403 > 530 [i]
- extracting embedded orthogonal array [i] would yield OA(530, 35, S5, 25), but