Best Known (96−81, 96, s)-Nets in Base 8
(96−81, 96, 65)-Net over F8 — Constructive and digital
Digital (15, 96, 65)-net over F8, using
- t-expansion [i] based on digital (14, 96, 65)-net over F8, using
- net from sequence [i] based on digital (14, 64)-sequence over F8, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 14 and N(F) ≥ 65, using
- the Suzuki function field over F8 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 14 and N(F) ≥ 65, using
- net from sequence [i] based on digital (14, 64)-sequence over F8, using
(96−81, 96, 271)-Net over F8 — Upper bound on s (digital)
There is no digital (15, 96, 272)-net over F8, because
- 1 times m-reduction [i] would yield digital (15, 95, 272)-net over F8, but
- extracting embedded orthogonal array [i] would yield linear OA(895, 272, F8, 80) (dual of [272, 177, 81]-code), but
- residual code [i] would yield OA(815, 191, S8, 10), but
- the linear programming bound shows that M ≥ 25 427098 382149 761175 126016 / 717528 710051 > 815 [i]
- residual code [i] would yield OA(815, 191, S8, 10), but
- extracting embedded orthogonal array [i] would yield linear OA(895, 272, F8, 80) (dual of [272, 177, 81]-code), but
(96−81, 96, 288)-Net in Base 8 — Upper bound on s
There is no (15, 96, 289)-net in base 8, because
- 5 times m-reduction [i] would yield (15, 91, 289)-net in base 8, but
- the generalized Rao bound for nets shows that 8m ≥ 15612 322234 848331 880601 230359 684924 233263 993047 381795 871050 142694 117687 977369 078400 > 891 [i]