Best Known (69, 69+69, s)-Nets in Base 2
(69, 69+69, 48)-Net over F2 — Constructive and digital
Digital (69, 138, 48)-net over F2, using
- net from sequence [i] based on digital (69, 47)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 69 and N(F) ≥ 48, using
(69, 69+69, 49)-Net over F2 — Digital
Digital (69, 138, 49)-net over F2, using
- t-expansion [i] based on digital (68, 138, 49)-net over F2, using
- net from sequence [i] based on digital (68, 48)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 68 and N(F) ≥ 49, using
- net from sequence [i] based on digital (68, 48)-sequence over F2, using
(69, 69+69, 152)-Net over F2 — Upper bound on s (digital)
There is no digital (69, 138, 153)-net over F2, because
- 1 times m-reduction [i] would yield digital (69, 137, 153)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2137, 153, F2, 68) (dual of [153, 16, 69]-code), but
- residual code [i] would yield linear OA(269, 84, F2, 34) (dual of [84, 15, 35]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(2137, 153, F2, 68) (dual of [153, 16, 69]-code), but
(69, 69+69, 175)-Net in Base 2 — Upper bound on s
There is no (69, 138, 176)-net in base 2, because
- 1 times m-reduction [i] would yield (69, 137, 176)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 196792 741687 243045 673614 634077 952520 356793 > 2137 [i]