Best Known (41−27, 41, s)-Nets in Base 3
(41−27, 41, 24)-Net over F3 — Constructive and digital
Digital (14, 41, 24)-net over F3, using
- t-expansion [i] based on digital (13, 41, 24)-net over F3, using
- net from sequence [i] based on digital (13, 23)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 13 and N(F) ≥ 24, using
- net from sequence [i] based on digital (13, 23)-sequence over F3, using
(41−27, 41, 54)-Net over F3 — Upper bound on s (digital)
There is no digital (14, 41, 55)-net over F3, because
- extracting embedded orthogonal array [i] would yield linear OA(341, 55, F3, 27) (dual of [55, 14, 28]-code), but
- construction Y1 [i] would yield
- linear OA(340, 47, F3, 27) (dual of [47, 7, 28]-code), but
- construction Y1 [i] would yield
- linear OA(339, 43, F3, 27) (dual of [43, 4, 28]-code), but
- OA(37, 47, S3, 4), but
- discarding factors would yield OA(37, 34, S3, 4), but
- the Rao or (dual) Hamming bound shows that M ≥ 2313 > 37 [i]
- discarding factors would yield OA(37, 34, S3, 4), but
- construction Y1 [i] would yield
- OA(314, 55, S3, 8), but
- discarding factors would yield OA(314, 53, S3, 8), but
- the Rao or (dual) Hamming bound shows that M ≥ 4 878227 > 314 [i]
- discarding factors would yield OA(314, 53, S3, 8), but
- linear OA(340, 47, F3, 27) (dual of [47, 7, 28]-code), but
- construction Y1 [i] would yield
(41−27, 41, 59)-Net in Base 3 — Upper bound on s
There is no (14, 41, 60)-net in base 3, because
- extracting embedded orthogonal array [i] would yield OA(341, 60, S3, 27), but
- the linear programming bound shows that M ≥ 116 204898 843079 582412 588334 166989 / 3 151520 781409 > 341 [i]