Best Known (130−63, 130, s)-Nets in Base 2
(130−63, 130, 43)-Net over F2 — Constructive and digital
Digital (67, 130, 43)-net over F2, using
- t-expansion [i] based on digital (59, 130, 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
(130−63, 130, 48)-Net over F2 — Digital
Digital (67, 130, 48)-net over F2, using
- t-expansion [i] based on digital (65, 130, 48)-net over F2, using
- net from sequence [i] based on digital (65, 47)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 65 and N(F) ≥ 48, using
- net from sequence [i] based on digital (65, 47)-sequence over F2, using
(130−63, 130, 151)-Net over F2 — Upper bound on s (digital)
There is no digital (67, 130, 152)-net over F2, because
- 1 times m-reduction [i] would yield digital (67, 129, 152)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2129, 152, F2, 62) (dual of [152, 23, 63]-code), but
- 1 times code embedding in larger space [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
- 1 times code embedding in larger space [i] would yield linear OA(2130, 153, F2, 62) (dual of [153, 23, 63]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(2129, 152, F2, 62) (dual of [152, 23, 63]-code), but
(130−63, 130, 180)-Net in Base 2 — Upper bound on s
There is no (67, 130, 181)-net in base 2, because
- 1 times m-reduction [i] would yield (67, 129, 181)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 773 901620 193970 258084 211259 472490 874464 > 2129 [i]