Best Known (65−32, 65, s)-Nets in Base 2
(65−32, 65, 24)-Net over F2 — Constructive and digital
Digital (33, 65, 24)-net over F2, using
- net from sequence [i] based on digital (33, 23)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 33 and N(F) ≥ 24, using
(65−32, 65, 28)-Net over F2 — Digital
Digital (33, 65, 28)-net over F2, using
- net from sequence [i] based on digital (33, 27)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 33 and N(F) ≥ 28, using
(65−32, 65, 75)-Net over F2 — Upper bound on s (digital)
There is no digital (33, 65, 76)-net over F2, because
- extracting embedded orthogonal array [i] would yield linear OA(265, 76, F2, 32) (dual of [76, 11, 33]-code), but
- construction Y1 [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
- OA(211, 76, S2, 4), but
- discarding factors would yield OA(211, 64, S2, 4), but
- the Rao or (dual) Hamming bound shows that M ≥ 2081 > 211 [i]
- discarding factors would yield OA(211, 64, S2, 4), but
- linear OA(264, 72, F2, 32) (dual of [72, 8, 33]-code), but
- construction Y1 [i] would yield
(65−32, 65, 80)-Net in Base 2 — Upper bound on s
There is no (33, 65, 81)-net in base 2, because
- extracting embedded orthogonal array [i] would yield OA(265, 81, S2, 32), but
- the linear programming bound shows that M ≥ 648144 799773 858805 579776 / 14875 > 265 [i]