Best Known (65, 65+64, s)-Nets in Base 2
(65, 65+64, 43)-Net over F2 — Constructive and digital
Digital (65, 129, 43)-net over F2, using
- t-expansion [i] based on digital (59, 129, 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
(65, 65+64, 48)-Net over F2 — Digital
Digital (65, 129, 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
(65, 65+64, 140)-Net over F2 — Upper bound on s (digital)
There is no digital (65, 129, 141)-net over F2, because
- extracting embedded orthogonal array [i] would yield linear OA(2129, 141, F2, 64) (dual of [141, 12, 65]-code), but
- construction Y1 [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
- OA(212, 141, S2, 4), but
- discarding factors would yield OA(212, 91, S2, 4), but
- the Rao or (dual) Hamming bound shows that M ≥ 4187 > 212 [i]
- discarding factors would yield OA(212, 91, S2, 4), but
- linear OA(2128, 137, F2, 64) (dual of [137, 9, 65]-code), but
- construction Y1 [i] would yield
(65, 65+64, 143)-Net in Base 2 — Upper bound on s
There is no (65, 129, 144)-net in base 2, because
- 2 times m-reduction [i] would yield (65, 127, 144)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2127, 144, S2, 62), but
- the linear programming bound shows that M ≥ 1 097070 350953 105606 205919 734360 020713 734144 / 5719 > 2127 [i]
- extracting embedded orthogonal array [i] would yield OA(2127, 144, S2, 62), but