Best Known (139−75, 139, s)-Nets in Base 2
(139−75, 139, 43)-Net over F2 — Constructive and digital
Digital (64, 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
(139−75, 139, 44)-Net over F2 — Digital
Digital (64, 139, 44)-net over F2, using
- t-expansion [i] based on digital (62, 139, 44)-net over F2, using
- net from sequence [i] based on digital (62, 43)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 62 and N(F) ≥ 44, using
- net from sequence [i] based on digital (62, 43)-sequence over F2, using
(139−75, 139, 136)-Net over F2 — Upper bound on s (digital)
There is no digital (64, 139, 137)-net over F2, because
- 11 times m-reduction [i] would yield digital (64, 128, 137)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2128, 137, F2, 64) (dual of [137, 9, 65]-code), but
- residual code [i] would yield linear OA(264, 72, F2, 32) (dual of [72, 8, 33]-code), but
- adding a parity check bit [i] would yield linear OA(265, 73, F2, 33) (dual of [73, 8, 34]-code), but
- “BJV†bound on codes from Brouwer’s database [i]
- adding a parity check bit [i] would yield linear OA(265, 73, F2, 33) (dual of [73, 8, 34]-code), but
- residual code [i] would yield linear OA(264, 72, F2, 32) (dual of [72, 8, 33]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(2128, 137, F2, 64) (dual of [137, 9, 65]-code), but
(139−75, 139, 139)-Net in Base 2 — Upper bound on s
There is no (64, 139, 140)-net in base 2, because
- 11 times m-reduction [i] would yield (64, 128, 140)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2128, 140, S2, 64), but
- the linear programming bound shows that M ≥ 174224 571863 520493 293247 799005 065324 265472 / 429 > 2128 [i]
- extracting embedded orthogonal array [i] would yield OA(2128, 140, S2, 64), but