Best Known (3, 57, s)-Nets in Base 16
(3, 57, 38)-Net over F16 — Constructive and digital
Digital (3, 57, 38)-net over F16, using
- net from sequence [i] based on digital (3, 37)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 3 and N(F) ≥ 38, using
(3, 57, 67)-Net over F16 — Upper bound on s (digital)
There is no digital (3, 57, 68)-net over F16, because
- 6 times m-reduction [i] would yield digital (3, 51, 68)-net over F16, but
- extracting embedded orthogonal array [i] would yield linear OA(1651, 68, F16, 48) (dual of [68, 17, 49]-code), but
- residual code [i] would yield OA(163, 19, S16, 3), but
- 1 times truncation [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]
- 1 times truncation [i] would yield OA(162, 18, S16, 2), but
- residual code [i] would yield OA(163, 19, S16, 3), but
- extracting embedded orthogonal array [i] would yield linear OA(1651, 68, F16, 48) (dual of [68, 17, 49]-code), but
(3, 57, 82)-Net in Base 16 — Upper bound on s
There is no (3, 57, 83)-net in base 16, because
- extracting embedded orthogonal array [i] would yield OA(1657, 83, S16, 54), but
- the linear programming bound shows that M ≥ 750 378995 432047 752626 559807 149297 542184 811987 458850 712171 611389 957955 845108 006912 / 1 738489 690817 > 1657 [i]