Best Known (18−17, 18, s)-Nets in Base 16
(18−17, 18, 24)-Net over F16 — Constructive and digital
Digital (1, 18, 24)-net over F16, using
- net from sequence [i] based on digital (1, 23)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 1 and N(F) ≥ 24, using
(18−17, 18, 25)-Net over F16 — Digital
Digital (1, 18, 25)-net over F16, using
- net from sequence [i] based on digital (1, 24)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 1 and N(F) ≥ 25, using
(18−17, 18, 47)-Net over F16 — Upper bound on s (digital)
There is no digital (1, 18, 48)-net over F16, because
- 1 times m-reduction [i] would yield digital (1, 17, 48)-net over F16, but
- extracting embedded orthogonal array [i] would yield linear OA(1617, 48, F16, 16) (dual of [48, 31, 17]-code), but
- dual of a near-MDS code is again a near-MDS code [i] would yield linear OA(1631, 48, F16, 30) (dual of [48, 17, 31]-code), but
- discarding factors / shortening the dual code would yield linear OA(1631, 34, F16, 30) (dual of [34, 3, 31]-code), but
- dual of a near-MDS code is again a near-MDS code [i] would yield linear OA(1631, 48, F16, 30) (dual of [48, 17, 31]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(1617, 48, F16, 16) (dual of [48, 31, 17]-code), but
(18−17, 18, 73)-Net in Base 16 — Upper bound on s
There is no (1, 18, 74)-net in base 16, because
- 9 times m-reduction [i] would yield (1, 9, 74)-net in base 16, but
- the generalized Rao bound for nets shows that 16m ≥ 69462 347566 > 169 [i]
- extracting embedded orthogonal array [i] would yield OA(169, 74, S16, 8), but
- the linear programming bound shows that M ≥ 108 985717 478079 332352 / 1552 440097 > 169 [i]