Best Known (71−67, 71, s)-Nets in Base 16
(71−67, 71, 45)-Net over F16 — Constructive and digital
Digital (4, 71, 45)-net over F16, using
- net from sequence [i] based on digital (4, 44)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 4 and N(F) ≥ 45, using
(71−67, 71, 76)-Net over F16 — Upper bound on s (digital)
There is no digital (4, 71, 77)-net over F16, because
- 3 times m-reduction [i] would yield digital (4, 68, 77)-net over F16, but
- extracting embedded orthogonal array [i] would yield linear OA(1668, 77, F16, 64) (dual of [77, 9, 65]-code), but
- construction Y1 [i] would yield
- OA(1667, 69, S16, 64), but
- the (dual) Plotkin bound shows that M ≥ 7588 550360 256754 183279 148073 529370 729071 901715 047420 004889 892225 542594 864082 845696 / 13 > 1667 [i]
- OA(169, 77, S16, 8), but
- the Rao or (dual) Hamming bound shows that M ≥ 68757 087631 > 169 [i]
- OA(1667, 69, S16, 64), but
- construction Y1 [i] would yield
- extracting embedded orthogonal array [i] would yield linear OA(1668, 77, F16, 64) (dual of [77, 9, 65]-code), but
(71−67, 71, 107)-Net in Base 16 — Upper bound on s
There is no (4, 71, 108)-net in base 16, because
- extracting embedded orthogonal array [i] would yield OA(1671, 108, S16, 67), but
- the linear programming bound shows that M ≥ 5870 116485 270291 500967 861651 075763 626920 000025 245105 415735 178636 219956 916494 249448 823400 711845 838848 / 183 754725 749883 > 1671 [i]