Best Known (52−46, 52, s)-Nets in Base 9
(52−46, 52, 34)-Net over F9 — Constructive and digital
Digital (6, 52, 34)-net over F9, using
- net from sequence [i] based on digital (6, 33)-sequence over F9, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 6 and N(F) ≥ 34, using
(52−46, 52, 35)-Net over F9 — Digital
Digital (6, 52, 35)-net over F9, using
- net from sequence [i] based on digital (6, 34)-sequence over F9, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 6 and N(F) ≥ 35, using
(52−46, 52, 85)-Net over F9 — Upper bound on s (digital)
There is no digital (6, 52, 86)-net over F9, because
- 1 times m-reduction [i] would yield digital (6, 51, 86)-net over F9, but
- extracting embedded orthogonal array [i] would yield linear OA(951, 86, F9, 45) (dual of [86, 35, 46]-code), but
- construction Y1 [i] would yield
- linear OA(950, 55, F9, 45) (dual of [55, 5, 46]-code), but
- construction Y1 [i] would yield
- OA(949, 51, S9, 45), but
- the (dual) Plotkin bound shows that M ≥ 1 546132 562196 033993 109383 389296 863818 106322 566003 / 23 > 949 [i]
- OA(95, 55, S9, 4), but
- discarding factors would yield OA(95, 44, S9, 4), but
- the Rao or (dual) Hamming bound shows that M ≥ 60897 > 95 [i]
- discarding factors would yield OA(95, 44, S9, 4), but
- OA(949, 51, S9, 45), but
- construction Y1 [i] would yield
- OA(935, 86, S9, 31), but
- discarding factors would yield OA(935, 85, S9, 31), but
- the linear programming bound shows that M ≥ 112620 123087 737061 231244 499443 215625 747636 066957 184847 246285 / 44 721454 781644 057928 595589 > 935 [i]
- discarding factors would yield OA(935, 85, S9, 31), but
- linear OA(950, 55, F9, 45) (dual of [55, 5, 46]-code), but
- construction Y1 [i] would yield
- extracting embedded orthogonal array [i] would yield linear OA(951, 86, F9, 45) (dual of [86, 35, 46]-code), but
(52−46, 52, 94)-Net in Base 9 — Upper bound on s
There is no (6, 52, 95)-net in base 9, because
- extracting embedded orthogonal array [i] would yield OA(952, 95, S9, 46), but
- the linear programming bound shows that M ≥ 156 594191 819757 241614 916139 836305 409895 623917 021547 780625 517695 115139 666658 176280 301351 / 3 716465 190571 598442 739998 255746 300023 > 952 [i]