Best Known (68, 131, s)-Nets in Base 2
(68, 131, 43)-Net over F2 — Constructive and digital
Digital (68, 131, 43)-net over F2, using
- t-expansion [i] based on digital (59, 131, 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
(68, 131, 49)-Net over F2 — Digital
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
(68, 131, 152)-Net over F2 — Upper bound on s (digital)
There is no digital (68, 131, 153)-net over F2, because
- 1 times m-reduction [i] would yield digital (68, 130, 153)-net over F2, but
- extracting embedded orthogonal array [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
- extracting embedded orthogonal array [i] would yield linear OA(2130, 153, F2, 62) (dual of [153, 23, 63]-code), but
(68, 131, 185)-Net in Base 2 — Upper bound on s
There is no (68, 131, 186)-net in base 2, because
- 1 times m-reduction [i] would yield (68, 130, 186)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 1554 917562 487597 233774 231792 581847 227952 > 2130 [i]