Best Known (57−55, 57, s)-Nets in Base 25
(57−55, 57, 28)-Net over F25 — Constructive and digital
Digital (2, 57, 28)-net over F25, using
- net from sequence [i] based on digital (2, 27)-sequence over F25, using
(57−55, 57, 46)-Net over F25 — Digital
Digital (2, 57, 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
(57−55, 57, 77)-Net over F25 — Upper bound on s (digital)
There is no digital (2, 57, 78)-net over F25, because
- 5 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
(57−55, 57, 151)-Net in Base 25 — Upper bound on s
There is no (2, 57, 152)-net in base 25, because
- extracting embedded orthogonal array [i] would yield OA(2557, 152, S25, 55), but
- the linear programming bound shows that M ≥ 13 721148 017699 488917 584122 781548 600823 898704 032367 560300 652298 277412 910465 500317 513942 718505 859375 / 277330 026760 748083 > 2557 [i]