Best Known (53−47, 53, s)-Nets in Base 9
(53−47, 53, 34)-Net over F9 — Constructive and digital
Digital (6, 53, 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
(53−47, 53, 35)-Net over F9 — Digital
Digital (6, 53, 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
(53−47, 53, 85)-Net over F9 — Upper bound on s (digital)
There is no digital (6, 53, 86)-net over F9, because
- 2 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
(53−47, 53, 90)-Net in Base 9 — Upper bound on s
There is no (6, 53, 91)-net in base 9, because
- extracting embedded orthogonal array [i] would yield OA(953, 91, S9, 47), but
- the linear programming bound shows that M ≥ 1 805866 253412 423561 017231 691027 796883 544324 423036 889068 209415 907113 599556 237288 179883 / 4654 018646 492649 551136 876628 773107 > 953 [i]