Best Known (109−106, 109, s)-Nets in Base 32
(109−106, 109, 64)-Net over F32 — Constructive and digital
Digital (3, 109, 64)-net over F32, using
- net from sequence [i] based on digital (3, 63)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 3 and N(F) ≥ 64, using
(109−106, 109, 131)-Net over F32 — Upper bound on s (digital)
There is no digital (3, 109, 132)-net over F32, because
- 10 times m-reduction [i] would yield digital (3, 99, 132)-net over F32, but
- extracting embedded orthogonal array [i] would yield linear OA(3299, 132, F32, 96) (dual of [132, 33, 97]-code), but
- residual code [i] would yield OA(323, 35, S32, 3), but
- 1 times truncation [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]
- 1 times truncation [i] would yield OA(322, 34, S32, 2), but
- residual code [i] would yield OA(323, 35, S32, 3), but
- extracting embedded orthogonal array [i] would yield linear OA(3299, 132, F32, 96) (dual of [132, 33, 97]-code), but
(109−106, 109, 224)-Net in Base 32 — Upper bound on s
There is no (3, 109, 225)-net in base 32, because
- 9 times m-reduction [i] would yield (3, 100, 225)-net in base 32, but
- extracting embedded orthogonal array [i] would yield OA(32100, 225, S32, 97), but
- the linear programming bound shows that M ≥ 2 176621 357211 405922 874416 376296 464536 271980 950849 045345 539822 984706 307743 412210 042179 441204 168005 917650 563663 281948 862935 325493 948854 158937 218572 505987 463250 532393 307695 218688 / 638810 956324 083601 474063 > 32100 [i]
- extracting embedded orthogonal array [i] would yield OA(32100, 225, S32, 97), but