Best Known (84−82, 84, s)-Nets in Base 32
(84−82, 84, 44)-Net over F32 — Constructive and digital
Digital (2, 84, 44)-net over F32, using
- t-expansion [i] based on digital (1, 84, 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
(84−82, 84, 53)-Net over F32 — Digital
Digital (2, 84, 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
(84−82, 84, 98)-Net over F32 — Upper bound on s (digital)
There is no digital (2, 84, 99)-net over F32, because
- 18 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
(84−82, 84, 171)-Net in Base 32 — Upper bound on s
There is no (2, 84, 172)-net in base 32, because
- extracting embedded orthogonal array [i] would yield OA(3284, 172, S32, 82), but
- the linear programming bound shows that M ≥ 55247 037733 293434 061032 503217 217874 783879 875974 189849 376042 794293 976190 858815 670524 653995 680660 403017 825779 663068 648234 977192 570295 156736 / 20118 640571 > 3284 [i]