Best Known (45−30, 45, s)-Nets in Base 3
(45−30, 45, 28)-Net over F3 — Constructive and digital
Digital (15, 45, 28)-net over F3, using
- net from sequence [i] based on digital (15, 27)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 15 and N(F) ≥ 28, using
(45−30, 45, 55)-Net over F3 — Upper bound on s (digital)
There is no digital (15, 45, 56)-net over F3, because
- extracting embedded orthogonal array [i] would yield linear OA(345, 56, F3, 30) (dual of [56, 11, 31]-code), but
- construction Y1 [i] would yield
- linear OA(344, 50, F3, 30) (dual of [50, 6, 31]-code), but
- “HJL†bound on codes from Brouwer’s database [i]
- OA(311, 56, S3, 6), but
- discarding factors would yield OA(311, 52, S3, 6), but
- the Rao or (dual) Hamming bound shows that M ≥ 182209 > 311 [i]
- discarding factors would yield OA(311, 52, S3, 6), but
- linear OA(344, 50, F3, 30) (dual of [50, 6, 31]-code), but
- construction Y1 [i] would yield
(45−30, 45, 58)-Net in Base 3 — Upper bound on s
There is no (15, 45, 59)-net in base 3, because
- extracting embedded orthogonal array [i] would yield OA(345, 59, S3, 30), but
- the linear programming bound shows that M ≥ 15711 197460 636193 905236 899365 / 4 636174 > 345 [i]