Best Known (9−7, 9, s)-Nets in Base 7
(9−7, 9, 13)-Net over F7 — Constructive and digital
Digital (2, 9, 13)-net over F7, using
- t-expansion [i] based on digital (1, 9, 13)-net over F7, using
- 4 times m-reduction [i] based on digital (1, 13, 13)-net over F7, using
(9−7, 9, 16)-Net over F7 — Digital
Digital (2, 9, 16)-net over F7, using
- net from sequence [i] based on digital (2, 15)-sequence over F7, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F7 with g(F) = 2 and N(F) ≥ 16, using
(9−7, 9, 48)-Net over F7 — Upper bound on s (digital)
There is no digital (2, 9, 49)-net over F7, because
- extracting embedded orthogonal array [i] would yield linear OA(79, 49, F7, 7) (dual of [49, 40, 8]-code), but
- construction Y1 [i] would yield
- linear OA(78, 14, F7, 7) (dual of [14, 6, 8]-code), but
- “MPa†bound on codes from Brouwer’s database [i]
- OA(740, 49, S7, 35), but
- discarding factors would yield OA(740, 47, S7, 35), but
- the linear programming bound shows that M ≥ 721 282691 041861 962448 675704 052408 625288 / 101475 > 740 [i]
- discarding factors would yield OA(740, 47, S7, 35), but
- linear OA(78, 14, F7, 7) (dual of [14, 6, 8]-code), but
- construction Y1 [i] would yield
(9−7, 9, 52)-Net in Base 7 — Upper bound on s
There is no (2, 9, 53)-net in base 7, because
- 1 times m-reduction [i] would yield (2, 8, 53)-net in base 7, but
- the generalized Rao bound for nets shows that 7m ≥ 5 822263 > 78 [i]