Best Known (131−62, 131, s)-Nets in Base 2
(131−62, 131, 48)-Net over F2 — Constructive and digital
Digital (69, 131, 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
(131−62, 131, 49)-Net over F2 — Digital
Digital (69, 131, 49)-net over F2, using
- t-expansion [i] based on digital (68, 131, 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
(131−62, 131, 164)-Net over F2 — Upper bound on s (digital)
There is no digital (69, 131, 165)-net over F2, because
- extracting embedded orthogonal array [i] would yield linear OA(2131, 165, F2, 62) (dual of [165, 34, 63]-code), but
- construction Y1 [i] would yield
- linear OA(2130, 153, F2, 62) (dual of [153, 23, 63]-code), but
- adding a parity check bit [i] would yield linear OA(2131, 154, F2, 63) (dual of [154, 23, 64]-code), but
- OA(234, 165, S2, 12), but
- discarding factors would yield OA(234, 154, S2, 12), but
- the Rao or (dual) Hamming bound shows that M ≥ 17486 314616 > 234 [i]
- discarding factors would yield OA(234, 154, S2, 12), but
- linear OA(2130, 153, F2, 62) (dual of [153, 23, 63]-code), but
- construction Y1 [i] would yield
(131−62, 131, 190)-Net in Base 2 — Upper bound on s
There is no (69, 131, 191)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 3075 219080 269104 775098 980815 069678 447050 > 2131 [i]