Best Known (2, 41, s)-Nets in Base 16
(2, 41, 33)-Net over F16 — Constructive and digital
Digital (2, 41, 33)-net over F16, using
- net from sequence [i] based on digital (2, 32)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 2 and N(F) ≥ 33, using
(2, 41, 50)-Net over F16 — Upper bound on s (digital)
There is no digital (2, 41, 51)-net over F16, because
- 7 times m-reduction [i] would yield digital (2, 34, 51)-net over F16, but
- extracting embedded orthogonal array [i] would yield linear OA(1634, 51, F16, 32) (dual of [51, 17, 33]-code), but
- residual code [i] would yield OA(162, 18, S16, 2), but
- bound for OAs with strength k = 2 [i]
- the Rao or (dual) Hamming bound shows that M ≥ 271 > 162 [i]
- residual code [i] would yield OA(162, 18, S16, 2), but
- extracting embedded orthogonal array [i] would yield linear OA(1634, 51, F16, 32) (dual of [51, 17, 33]-code), but
(2, 41, 88)-Net in Base 16 — Upper bound on s
There is no (2, 41, 89)-net in base 16, because
- extracting embedded orthogonal array [i] would yield OA(1641, 89, S16, 39), but
- the linear programming bound shows that M ≥ 198597 513856 191693 689387 369499 523269 630395 204439 919135 031296 / 8302 050811 > 1641 [i]