Best Known (68, 68+71, s)-Nets in Base 2
(68, 68+71, 43)-Net over F2 — Constructive and digital
Digital (68, 139, 43)-net over F2, using
- t-expansion [i] based on digital (59, 139, 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, 68+71, 49)-Net over F2 — Digital
Digital (68, 139, 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, 68+71, 146)-Net over F2 — Upper bound on s (digital)
There is no digital (68, 139, 147)-net over F2, because
- 1 times m-reduction [i] would yield digital (68, 138, 147)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2138, 147, F2, 70) (dual of [147, 9, 71]-code), but
- residual code [i] would yield linear OA(268, 76, F2, 35) (dual of [76, 8, 36]-code), but
- “BJV†bound on codes from Brouwer’s database [i]
- residual code [i] would yield linear OA(268, 76, F2, 35) (dual of [76, 8, 36]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(2138, 147, F2, 70) (dual of [147, 9, 71]-code), but
(68, 68+71, 147)-Net in Base 2 — Upper bound on s
There is no (68, 139, 148)-net in base 2, because
- 3 times m-reduction [i] would yield (68, 136, 148)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2136, 148, S2, 68), but
- the linear programming bound shows that M ≥ 646 721610 757388 071104 535829 906802 483673 432064 / 6027 > 2136 [i]
- extracting embedded orthogonal array [i] would yield OA(2136, 148, S2, 68), but