Best Known (8, 8+70, s)-Nets in Base 7
(8, 8+70, 16)-Net over F7 — Constructive and digital
Digital (8, 78, 16)-net over F7, using
- net from sequence [i] based on digital (8, 15)-sequence over F7, using
(8, 8+70, 32)-Net over F7 — Digital
Digital (8, 78, 32)-net over F7, using
- t-expansion [i] based on digital (7, 78, 32)-net over F7, using
- net from sequence [i] based on digital (7, 31)-sequence over F7, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F7 with g(F) = 7 and N(F) ≥ 32, using
- net from sequence [i] based on digital (7, 31)-sequence over F7, using
(8, 8+70, 63)-Net over F7 — Upper bound on s (digital)
There is no digital (8, 78, 64)-net over F7, because
- 21 times m-reduction [i] would yield digital (8, 57, 64)-net over F7, but
- extracting embedded orthogonal array [i] would yield linear OA(757, 64, F7, 49) (dual of [64, 7, 50]-code), but
- residual code [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]
- residual code [i] would yield linear OA(78, 14, F7, 7) (dual of [14, 6, 8]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(757, 64, F7, 49) (dual of [64, 7, 50]-code), but
(8, 8+70, 66)-Net in Base 7 — Upper bound on s
There is no (8, 78, 67)-net in base 7, because
- 21 times m-reduction [i] would yield (8, 57, 67)-net in base 7, but
- extracting embedded orthogonal array [i] would yield OA(757, 67, S7, 49), but
- the linear programming bound shows that M ≥ 14 690613 296110 757318 840379 593851 330929 325424 135975 451203 / 9 440000 > 757 [i]
- extracting embedded orthogonal array [i] would yield OA(757, 67, S7, 49), but