Best Known (18−16, 18, s)-Nets in Base 8
(18−16, 18, 17)-Net over F8 — Constructive and digital
Digital (2, 18, 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
(18−16, 18, 18)-Net over F8 — Digital
Digital (2, 18, 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
(18−16, 18, 26)-Net over F8 — Upper bound on s (digital)
There is no digital (2, 18, 27)-net over F8, because
- 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
(18−16, 18, 50)-Net in Base 8 — Upper bound on s
There is no (2, 18, 51)-net in base 8, because
- extracting embedded orthogonal array [i] would yield OA(818, 51, S8, 16), but
- the linear programming bound shows that M ≥ 7 011863 238134 834281 840640 / 376 538367 > 818 [i]