Best Known (65−63, 65, s)-Nets in Base 25
(65−63, 65, 28)-Net over F25 — Constructive and digital
Digital (2, 65, 28)-net over F25, using
- net from sequence [i] based on digital (2, 27)-sequence over F25, using
(65−63, 65, 46)-Net over F25 — Digital
Digital (2, 65, 46)-net over F25, using
- net from sequence [i] based on digital (2, 45)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 2 and N(F) ≥ 46, using
(65−63, 65, 77)-Net over F25 — Upper bound on s (digital)
There is no digital (2, 65, 78)-net over F25, because
- 13 times m-reduction [i] would yield digital (2, 52, 78)-net over F25, but
- extracting embedded orthogonal array [i] would yield linear OA(2552, 78, F25, 50) (dual of [78, 26, 51]-code), but
- residual code [i] would yield OA(252, 27, S25, 2), but
- bound for OAs with strength k = 2 [i]
- the Rao or (dual) Hamming bound shows that M ≥ 649 > 252 [i]
- residual code [i] would yield OA(252, 27, S25, 2), but
- extracting embedded orthogonal array [i] would yield linear OA(2552, 78, F25, 50) (dual of [78, 26, 51]-code), but
(65−63, 65, 134)-Net in Base 25 — Upper bound on s
There is no (2, 65, 135)-net in base 25, because
- extracting embedded orthogonal array [i] would yield OA(2565, 135, S25, 63), but
- the linear programming bound shows that M ≥ 8535 453508 159081 511831 574294 410514 561385 359335 510316 175174 051755 991655 454636 202193 796634 674072 265625 / 1158 116937 > 2565 [i]