Best Known (1, ∞, s)-Nets in Base 25
(1, ∞, 27)-Net over F25 — Constructive and digital
Digital (1, m, 27)-net over F25 for arbitrarily large m, using
- net from sequence [i] based on digital (1, 26)-sequence over F25, using
(1, ∞, 36)-Net over F25 — Digital
Digital (1, m, 36)-net over F25 for arbitrarily large m, using
- net from sequence [i] based on digital (1, 35)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 1 and N(F) ≥ 36, using
(1, ∞, 51)-Net over F25 — Upper bound on s (digital)
There is no digital (1, m, 52)-net over F25 for arbitrarily large m, because
- m-reduction [i] would yield digital (1, 49, 52)-net over F25, but
- extracting embedded orthogonal array [i] would yield linear OA(2549, 52, F25, 48) (dual of [52, 3, 49]-code), but
(1, ∞, 52)-Net in Base 25 — Upper bound on s
There is no (1, m, 53)-net in base 25 for arbitrarily large m, because
- m-reduction [i] would yield (1, 51, 53)-net in base 25, but
- extracting embedded orthogonal array [i] would yield OA(2551, 53, S25, 50), but
- the (dual) Plotkin bound shows that M ≥ 4 930380 657631 323783 823303 533017 413935 457540 219431 393779 814243 316650 390625 / 17 > 2551 [i]
- extracting embedded orthogonal array [i] would yield OA(2551, 53, S25, 50), but