Best Known (68−64, 68, s)-Nets in Base 16
(68−64, 68, 45)-Net over F16 — Constructive and digital
Digital (4, 68, 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
(68−64, 68, 76)-Net over F16 — Upper bound on s (digital)
There is no digital (4, 68, 77)-net over F16, because
- 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
(68−64, 68, 119)-Net in Base 16 — Upper bound on s
There is no (4, 68, 120)-net in base 16, because
- extracting embedded orthogonal array [i] would yield OA(1668, 120, S16, 64), but
- the linear programming bound shows that M ≥ 64 355408 071103 283869 904172 583116 023781 438807 995516 085587 129269 676958 840251 368997 656824 564547 636587 437838 303232 / 8131 704336 230433 604612 971485 > 1668 [i]