Best Known (6, 6+36, s)-Nets in Base 7
(6, 6+36, 14)-Net over F7 — Constructive and digital
Digital (6, 42, 14)-net over F7, using
- net from sequence [i] based on digital (6, 13)-sequence over F7, using
(6, 6+36, 24)-Net over F7 — Digital
Digital (6, 42, 24)-net over F7, using
- t-expansion [i] based on digital (4, 42, 24)-net over F7, using
- net from sequence [i] based on digital (4, 23)-sequence over F7, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F7 with g(F) = 4 and N(F) ≥ 24, using
- net from sequence [i] based on digital (4, 23)-sequence over F7, using
(6, 6+36, 65)-Net over F7 — Upper bound on s (digital)
There is no digital (6, 42, 66)-net over F7, because
- 1 times m-reduction [i] would yield digital (6, 41, 66)-net over F7, but
- extracting embedded orthogonal array [i] would yield linear OA(741, 66, F7, 35) (dual of [66, 25, 36]-code), but
- residual code [i] would yield OA(76, 30, S7, 5), but
- 1 times truncation [i] would yield OA(75, 29, S7, 4), but
- the linear programming bound shows that M ≥ 3 384381 / 197 > 75 [i]
- 1 times truncation [i] would yield OA(75, 29, S7, 4), but
- residual code [i] would yield OA(76, 30, S7, 5), but
- extracting embedded orthogonal array [i] would yield linear OA(741, 66, F7, 35) (dual of [66, 25, 36]-code), but
(6, 6+36, 71)-Net in Base 7 — Upper bound on s
There is no (6, 42, 72)-net in base 7, because
- extracting embedded orthogonal array [i] would yield OA(742, 72, S7, 36), but
- the linear programming bound shows that M ≥ 1317 858742 541027 651897 283715 012422 855235 377901 886670 269689 / 4192 095566 060516 873025 > 742 [i]