Best Known (3, 3+45, s)-Nets in Base 16
(3, 3+45, 38)-Net over F16 — Constructive and digital
Digital (3, 48, 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, 3+45, 96)-Net over F16 — Upper bound on s (digital)
There is no digital (3, 48, 97)-net over F16, because
- 2 times m-reduction [i] would yield digital (3, 46, 97)-net over F16, but
- extracting embedded orthogonal array [i] would yield linear OA(1646, 97, F16, 43) (dual of [97, 51, 44]-code), but
- construction Y1 [i] would yield
- OA(1645, 49, S16, 43), but
- the linear programming bound shows that M ≥ 79 689768 125026 220634 634045 411816 077548 174434 353547 313152 / 47 > 1645 [i]
- linear OA(1651, 97, F16, 48) (dual of [97, 46, 49]-code), but
- discarding factors / shortening the dual code 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
- discarding factors / shortening the dual code would yield linear OA(1651, 68, F16, 48) (dual of [68, 17, 49]-code), but
- OA(1645, 49, S16, 43), but
- construction Y1 [i] would yield
- extracting embedded orthogonal array [i] would yield linear OA(1646, 97, F16, 43) (dual of [97, 51, 44]-code), but
(3, 3+45, 128)-Net in Base 16 — Upper bound on s
There is no (3, 48, 129)-net in base 16, because
- extracting embedded orthogonal array [i] would yield OA(1648, 129, S16, 45), but
- the linear programming bound shows that M ≥ 9 774115 429029 774558 670704 415804 096119 587642 027125 674755 540770 276947 209448 587264 / 1540 099078 102998 413999 > 1648 [i]