Best Known (72−67, 72, s)-Nets in Base 16
(72−67, 72, 49)-Net over F16 — Constructive and digital
Digital (5, 72, 49)-net over F16, using
- net from sequence [i] based on digital (5, 48)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 5 and N(F) ≥ 49, using
(72−67, 72, 158)-Net over F16 — Upper bound on s (digital)
There is no digital (5, 72, 159)-net over F16, because
- 3 times m-reduction [i] would yield digital (5, 69, 159)-net over F16, but
- extracting embedded orthogonal array [i] would yield linear OA(1669, 159, F16, 64) (dual of [159, 90, 65]-code), but
- residual code [i] would yield OA(165, 94, S16, 4), but
- the linear programming bound shows that M ≥ 2179 760128 / 2071 > 165 [i]
- residual code [i] would yield OA(165, 94, S16, 4), but
- extracting embedded orthogonal array [i] would yield linear OA(1669, 159, F16, 64) (dual of [159, 90, 65]-code), but
(72−67, 72, 173)-Net in Base 16 — Upper bound on s
There is no (5, 72, 174)-net in base 16, because
- extracting embedded orthogonal array [i] would yield OA(1672, 174, S16, 67), but
- the linear programming bound shows that M ≥ 560 737525 674747 363067 330815 928712 901261 158949 531870 757864 797378 782898 494202 744538 167813 056806 207624 896597 310302 385827 992043 378400 296960 / 1 094427 642440 264339 089389 600180 873522 835540 214959 > 1672 [i]