Best Known (14, 14+29, s)-Nets in Base 3
(14, 14+29, 24)-Net over F3 — Constructive and digital
Digital (14, 43, 24)-net over F3, using
- t-expansion [i] based on digital (13, 43, 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
(14, 14+29, 52)-Net over F3 — Upper bound on s (digital)
There is no digital (14, 43, 53)-net over F3, because
- 1 times m-reduction [i] would yield digital (14, 42, 53)-net over F3, but
- 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
- extracting embedded orthogonal array [i] would yield linear OA(342, 53, F3, 28) (dual of [53, 11, 29]-code), but
(14, 14+29, 55)-Net in Base 3 — Upper bound on s
There is no (14, 43, 56)-net in base 3, because
- extracting embedded orthogonal array [i] would yield OA(343, 56, S3, 29), but
- the linear programming bound shows that M ≥ 3 908555 710766 752983 304689 / 9635 > 343 [i]