Best Known (234−35, 234, s)-Nets in Base 2
(234−35, 234, 624)-Net over F2 — Constructive and digital
Digital (199, 234, 624)-net over F2, using
- t-expansion [i] based on digital (198, 234, 624)-net over F2, using
- trace code for nets [i] based on digital (3, 39, 104)-net over F64, using
- net from sequence [i] based on digital (3, 103)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 3 and N(F) ≥ 104, using
- net from sequence [i] based on digital (3, 103)-sequence over F64, using
- trace code for nets [i] based on digital (3, 39, 104)-net over F64, using
(234−35, 234, 2129)-Net over F2 — Digital
Digital (199, 234, 2129)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2234, 2129, F2, 3, 35) (dual of [(2129, 3), 6153, 36]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2234, 2746, F2, 3, 35) (dual of [(2746, 3), 8004, 36]-NRT-code), using
- OOA 3-folding [i] based on linear OA(2234, 8238, F2, 35) (dual of [8238, 8004, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(2234, 8240, F2, 35) (dual of [8240, 8006, 36]-code), using
- construction X applied to Ce(34) ⊂ Ce(28) [i] based on
- linear OA(2222, 8192, F2, 35) (dual of [8192, 7970, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(2183, 8192, F2, 29) (dual of [8192, 8009, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(212, 48, F2, 5) (dual of [48, 36, 6]-code), using
- adding a parity check bit [i] based on linear OA(211, 47, F2, 4) (dual of [47, 36, 5]-code), using
- extracting embedded orthogonal array [i] based on digital (7, 11, 47)-net over F2, using
- adding a parity check bit [i] based on linear OA(211, 47, F2, 4) (dual of [47, 36, 5]-code), using
- construction X applied to Ce(34) ⊂ Ce(28) [i] based on
- discarding factors / shortening the dual code based on linear OA(2234, 8240, F2, 35) (dual of [8240, 8006, 36]-code), using
- OOA 3-folding [i] based on linear OA(2234, 8238, F2, 35) (dual of [8238, 8004, 36]-code), using
- discarding factors / shortening the dual code based on linear OOA(2234, 2746, F2, 3, 35) (dual of [(2746, 3), 8004, 36]-NRT-code), using
(234−35, 234, 95876)-Net in Base 2 — Upper bound on s
There is no (199, 234, 95877)-net in base 2, because
- 1 times m-reduction [i] would yield (199, 233, 95877)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 13803 891918 804672 417171 809532 375789 516918 635695 120917 735847 678830 341584 > 2233 [i]