Best Known (42−29, 42, s)-Nets in Base 3
(42−29, 42, 24)-Net over F3 — Constructive and digital
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
(42−29, 42, 46)-Net over F3 — Upper bound on s (digital)
There is no digital (13, 42, 47)-net over F3, because
- 2 times m-reduction [i] would yield digital (13, 40, 47)-net over F3, but
- extracting embedded orthogonal array [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
- extracting embedded orthogonal array [i] would yield linear OA(340, 47, F3, 27) (dual of [47, 7, 28]-code), but
(42−29, 42, 48)-Net in Base 3 — Upper bound on s
There is no (13, 42, 49)-net in base 3, because
- extracting embedded orthogonal array [i] would yield OA(342, 49, S3, 29), but
- the linear programming bound shows that M ≥ 21664 959848 039447 123382 / 185 > 342 [i]