Best Known (2, 19, s)-Nets in Base 7
(2, 19, 10)-Net over F7 — Constructive and digital
Digital (2, 19, 10)-net over F7, using
- net from sequence [i] based on digital (2, 9)-sequence over F7, using
- Niederreiter sequence (Bratley–Fox–Niederreiter implementation) with equidistant coordinate [i]
(2, 19, 16)-Net over F7 — Digital
Digital (2, 19, 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
(2, 19, 23)-Net over F7 — Upper bound on s (digital)
There is no digital (2, 19, 24)-net over F7, because
- 3 times m-reduction [i] would yield digital (2, 16, 24)-net over F7, but
- extracting embedded orthogonal array [i] would yield linear OA(716, 24, F7, 14) (dual of [24, 8, 15]-code), but
- residual code [i] would yield OA(72, 9, S7, 2), but
- bound for OAs with strength k = 2 [i]
- the Rao or (dual) Hamming bound shows that M ≥ 55 > 72 [i]
- residual code [i] would yield OA(72, 9, S7, 2), but
- extracting embedded orthogonal array [i] would yield linear OA(716, 24, F7, 14) (dual of [24, 8, 15]-code), but
(2, 19, 39)-Net in Base 7 — Upper bound on s
There is no (2, 19, 40)-net in base 7, because
- extracting embedded orthogonal array [i] would yield OA(719, 40, S7, 17), but
- the linear programming bound shows that M ≥ 1 225528 563147 883820 373275 / 97 004732 > 719 [i]