Best Known (32−19, 32, s)-Nets in Base 2
(32−19, 32, 15)-Net over F2 — Constructive and digital
Digital (13, 32, 15)-net over F2, using
- net from sequence [i] based on digital (13, 14)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 13 and N(F) ≥ 15, using
- Niederreiter–Xing sequence (Piršić implementation) with equidistant coordinate [i]
(32−19, 32, 32)-Net over F2 — Upper bound on s (digital)
There is no digital (13, 32, 33)-net over F2, because
- 3 times m-reduction [i] would yield digital (13, 29, 33)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(229, 33, F2, 16) (dual of [33, 4, 17]-code), but
(32−19, 32, 33)-Net in Base 2 — Upper bound on s
There is no (13, 32, 34)-net in base 2, because
- 1 times m-reduction [i] would yield (13, 31, 34)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 2323 242356 > 231 [i]