Best Known (31−16, 31, s)-Nets in Base 2
(31−16, 31, 17)-Net over F2 — Constructive and digital
Digital (15, 31, 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
- Niederreiter–Xing sequence (Piršić implementation) with equidistant coordinate [i]
(31−16, 31, 37)-Net over F2 — Upper bound on s (digital)
There is no digital (15, 31, 38)-net over F2, because
- 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
(31−16, 31, 42)-Net in Base 2 — Upper bound on s
There is no (15, 31, 43)-net in base 2, because
- extracting embedded orthogonal array [i] would yield OA(231, 43, S2, 16), but
- the linear programming bound shows that M ≥ 2 430951 489536 / 1053 > 231 [i]