Best Known (127, 127+123, s)-Nets in Base 2
(127, 127+123, 57)-Net over F2 — Constructive and digital
Digital (127, 250, 57)-net over F2, using
- t-expansion [i] based on digital (110, 250, 57)-net over F2, using
- net from sequence [i] based on digital (110, 56)-sequence over F2, using
- Niederreiter–Xing sequence construction III based on the algebraic function field F/F2 with g(F) = 69, N(F) = 48, 1 place with degree 2, and 8 places with degree 6 [i] based on function field F/F2 with g(F) = 69 and N(F) ≥ 48, using an explicitly constructive algebraic function field [i]
- net from sequence [i] based on digital (110, 56)-sequence over F2, using
(127, 127+123, 81)-Net over F2 — Digital
Digital (127, 250, 81)-net over F2, using
- t-expansion [i] based on digital (126, 250, 81)-net over F2, using
- net from sequence [i] based on digital (126, 80)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 126 and N(F) ≥ 81, using
- net from sequence [i] based on digital (126, 80)-sequence over F2, using
(127, 127+123, 272)-Net over F2 — Upper bound on s (digital)
There is no digital (127, 250, 273)-net over F2, because
- 1 times m-reduction [i] would yield digital (127, 249, 273)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2249, 273, F2, 122) (dual of [273, 24, 123]-code), but
- residual code [i] would yield OA(2127, 150, S2, 61), but
- 1 times truncation [i] would yield OA(2126, 149, S2, 60), but
- the linear programming bound shows that M ≥ 40 354766 457887 934259 048521 444548 255732 989952 / 438495 > 2126 [i]
- 1 times truncation [i] would yield OA(2126, 149, S2, 60), but
- residual code [i] would yield OA(2127, 150, S2, 61), but
- extracting embedded orthogonal array [i] would yield linear OA(2249, 273, F2, 122) (dual of [273, 24, 123]-code), but
(127, 127+123, 277)-Net in Base 2 — Upper bound on s
There is no (127, 250, 278)-net in base 2, because
- 15 times m-reduction [i] would yield (127, 235, 278)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2235, 278, S2, 108), but
- the linear programming bound shows that M ≥ 8610 120597 997849 103535 796060 969471 925725 360797 699038 214930 323792 633970 252546 589383 983104 / 113493 110947 734375 > 2235 [i]
- extracting embedded orthogonal array [i] would yield OA(2235, 278, S2, 108), but