Best Known (78−76, 78, s)-Nets in Base 32
(78−76, 78, 44)-Net over F32 — Constructive and digital
Digital (2, 78, 44)-net over F32, using
- t-expansion [i] based on digital (1, 78, 44)-net over F32, using
- net from sequence [i] based on digital (1, 43)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 1 and N(F) ≥ 44, using
- net from sequence [i] based on digital (1, 43)-sequence over F32, using
(78−76, 78, 53)-Net over F32 — Digital
Digital (2, 78, 53)-net over F32, using
- net from sequence [i] based on digital (2, 52)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 2 and N(F) ≥ 53, using
(78−76, 78, 98)-Net over F32 — Upper bound on s (digital)
There is no digital (2, 78, 99)-net over F32, because
- 12 times m-reduction [i] would yield digital (2, 66, 99)-net over F32, but
- extracting embedded orthogonal array [i] would yield linear OA(3266, 99, F32, 64) (dual of [99, 33, 65]-code), but
- residual code [i] would yield OA(322, 34, S32, 2), but
- bound for OAs with strength k = 2 [i]
- the Rao or (dual) Hamming bound shows that M ≥ 1055 > 322 [i]
- residual code [i] would yield OA(322, 34, S32, 2), but
- extracting embedded orthogonal array [i] would yield linear OA(3266, 99, F32, 64) (dual of [99, 33, 65]-code), but
(78−76, 78, 179)-Net in Base 32 — Upper bound on s
There is no (2, 78, 180)-net in base 32, because
- extracting embedded orthogonal array [i] would yield OA(3278, 180, S32, 76), but
- the linear programming bound shows that M ≥ 1 833556 824999 389933 757633 231913 526357 704490 984079 585563 031471 536794 218642 391235 526611 060245 284343 606587 422297 648325 202726 570950 656000 / 722 337235 374959 > 3278 [i]