Best Known (247−37, 247, s)-Nets in Base 2
(247−37, 247, 624)-Net over F2 — Constructive and digital
Digital (210, 247, 624)-net over F2, using
- 21 times duplication [i] based on digital (209, 246, 624)-net over F2, using
- t-expansion [i] based on digital (208, 246, 624)-net over F2, using
- trace code for nets [i] based on digital (3, 41, 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, 41, 104)-net over F64, using
- t-expansion [i] based on digital (208, 246, 624)-net over F2, using
(247−37, 247, 2166)-Net over F2 — Digital
Digital (210, 247, 2166)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2247, 2166, F2, 3, 37) (dual of [(2166, 3), 6251, 38]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2247, 2746, F2, 3, 37) (dual of [(2746, 3), 7991, 38]-NRT-code), using
- OOA 3-folding [i] based on linear OA(2247, 8238, F2, 37) (dual of [8238, 7991, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(2247, 8240, F2, 37) (dual of [8240, 7993, 38]-code), using
- construction X applied to Ce(36) ⊂ Ce(30) [i] based on
- linear OA(2235, 8192, F2, 37) (dual of [8192, 7957, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(2196, 8192, F2, 31) (dual of [8192, 7996, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [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(36) ⊂ Ce(30) [i] based on
- discarding factors / shortening the dual code based on linear OA(2247, 8240, F2, 37) (dual of [8240, 7993, 38]-code), using
- OOA 3-folding [i] based on linear OA(2247, 8238, F2, 37) (dual of [8238, 7991, 38]-code), using
- discarding factors / shortening the dual code based on linear OOA(2247, 2746, F2, 3, 37) (dual of [(2746, 3), 7991, 38]-NRT-code), using
(247−37, 247, 98195)-Net in Base 2 — Upper bound on s
There is no (210, 247, 98196)-net in base 2, because
- 1 times m-reduction [i] would yield (210, 246, 98196)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 113 098391 655992 684625 330003 780646 381264 645980 552676 327409 111092 737440 874523 > 2246 [i]