Best Known (42−28, 42, s)-Nets in Base 3
(42−28, 42, 24)-Net over F3 — Constructive and digital
Digital (14, 42, 24)-net over F3, using
- t-expansion [i] based on digital (13, 42, 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
(42−28, 42, 52)-Net over F3 — Upper bound on s (digital)
There is no digital (14, 42, 53)-net over F3, because
- extracting embedded orthogonal array [i] would yield linear OA(342, 53, F3, 28) (dual of [53, 11, 29]-code), but
- construction Y1 [i] would yield
- linear OA(341, 47, F3, 28) (dual of [47, 6, 29]-code), but
- OA(311, 53, 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
- construction Y1 [i] would yield
(42−28, 42, 57)-Net in Base 3 — Upper bound on s
There is no (14, 42, 58)-net in base 3, because
- extracting embedded orthogonal array [i] would yield OA(342, 58, S3, 28), but
- the linear programming bound shows that M ≥ 16625 799496 955614 103330 196591 / 129 146875 > 342 [i]