Best Known (2, 21, s)-Nets in Base 8
(2, 21, 17)-Net over F8 — Constructive and digital
Digital (2, 21, 17)-net over F8, using
- net from sequence [i] based on digital (2, 16)-sequence over F8, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 2 and N(F) ≥ 17, using
(2, 21, 18)-Net over F8 — Digital
Digital (2, 21, 18)-net over F8, using
- net from sequence [i] based on digital (2, 17)-sequence over F8, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 2 and N(F) ≥ 18, using
(2, 21, 26)-Net over F8 — Upper bound on s (digital)
There is no digital (2, 21, 27)-net over F8, because
- 3 times m-reduction [i] would yield digital (2, 18, 27)-net over F8, but
- extracting embedded orthogonal array [i] would yield linear OA(818, 27, F8, 16) (dual of [27, 9, 17]-code), but
- residual code [i] would yield OA(82, 10, S8, 2), but
- bound for OAs with strength k = 2 [i]
- the Rao or (dual) Hamming bound shows that M ≥ 71 > 82 [i]
- residual code [i] would yield OA(82, 10, S8, 2), but
- extracting embedded orthogonal array [i] would yield linear OA(818, 27, F8, 16) (dual of [27, 9, 17]-code), but
(2, 21, 45)-Net in Base 8 — Upper bound on s
There is no (2, 21, 46)-net in base 8, because
- extracting embedded orthogonal array [i] would yield OA(821, 46, S8, 19), but
- the linear programming bound shows that M ≥ 14529 091252 314943 503823 011840 / 1479 210359 > 821 [i]