Best Known (135−72, 135, s)-Nets in Base 2
(135−72, 135, 43)-Net over F2 — Constructive and digital
Digital (63, 135, 43)-net over F2, using
- t-expansion [i] based on digital (59, 135, 43)-net over F2, using
- net from sequence [i] based on digital (59, 42)-sequence over F2, using
- Niederreiter–Xing sequence construction III based on the algebraic function field F/F2 with g(F) = 54, N(F) = 42, and 1 place with degree 6 [i] based on function field F/F2 with g(F) = 54 and N(F) ≥ 42, using an explicitly constructive algebraic function field [i]
- net from sequence [i] based on digital (59, 42)-sequence over F2, using
(135−72, 135, 44)-Net over F2 — Digital
Digital (63, 135, 44)-net over F2, using
- t-expansion [i] based on digital (62, 135, 44)-net over F2, using
- net from sequence [i] based on digital (62, 43)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 62 and N(F) ≥ 44, using
- net from sequence [i] based on digital (62, 43)-sequence over F2, using
(135−72, 135, 135)-Net over F2 — Upper bound on s (digital)
There is no digital (63, 135, 136)-net over F2, because
- 8 times m-reduction [i] would yield digital (63, 127, 136)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2127, 136, F2, 64) (dual of [136, 9, 65]-code), but
- residual code [i] would yield linear OA(263, 71, F2, 32) (dual of [71, 8, 33]-code), but
- residual code [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
- residual code [i] would yield linear OA(231, 38, F2, 16) (dual of [38, 7, 17]-code), but
- residual code [i] would yield linear OA(263, 71, F2, 32) (dual of [71, 8, 33]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(2127, 136, F2, 64) (dual of [136, 9, 65]-code), but
(135−72, 135, 137)-Net in Base 2 — Upper bound on s
There is no (63, 135, 138)-net in base 2, because
- 8 times m-reduction [i] would yield (63, 127, 138)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2127, 138, S2, 64), but
- the linear programming bound shows that M ≥ 246364 433650 759447 547483 215780 600185 094144 / 1287 > 2127 [i]
- extracting embedded orthogonal array [i] would yield OA(2127, 138, S2, 64), but