Best Known (15, 35, s)-Nets in Base 2
(15, 35, 17)-Net over F2 — Constructive and digital
Digital (15, 35, 17)-net over F2, using
- net from sequence [i] based on digital (15, 16)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 15 and N(F) ≥ 17, using
(15, 35, 37)-Net over F2 — Upper bound on s (digital)
There is no digital (15, 35, 38)-net over F2, because
- 4 times m-reduction [i] would yield digital (15, 31, 38)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(231, 38, F2, 16) (dual of [38, 7, 17]-code), but
- residual code [i] would yield linear OA(215, 21, F2, 8) (dual of [21, 6, 9]-code), but
- residual code [i] would yield linear OA(27, 12, F2, 4) (dual of [12, 5, 5]-code), but
- residual code [i] would yield linear OA(215, 21, F2, 8) (dual of [21, 6, 9]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(231, 38, F2, 16) (dual of [38, 7, 17]-code), but
(15, 35, 38)-Net in Base 2 — Upper bound on s
There is no (15, 35, 39)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 39104 534028 > 235 [i]
- extracting embedded orthogonal array [i] would yield OA(235, 39, S2, 20), but
- the (dual) Plotkin bound shows that M ≥ 274877 906944 / 7 > 235 [i]