Best Known (244−33, 244, s)-Nets in Base 2
(244−33, 244, 2049)-Net over F2 — Constructive and digital
Digital (211, 244, 2049)-net over F2, using
- 21 times duplication [i] based on digital (210, 243, 2049)-net over F2, using
- net defined by OOA [i] based on linear OOA(2243, 2049, F2, 33, 33) (dual of [(2049, 33), 67374, 34]-NRT-code), using
- OOA 16-folding and stacking with additional row [i] based on linear OA(2243, 32785, F2, 33) (dual of [32785, 32542, 34]-code), using
- 1 times code embedding in larger space [i] based on linear OA(2242, 32784, F2, 33) (dual of [32784, 32542, 34]-code), using
- construction X applied to Ce(32) ⊂ Ce(30) [i] based on
- linear OA(2241, 32768, F2, 33) (dual of [32768, 32527, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(2226, 32768, F2, 31) (dual of [32768, 32542, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(21, 16, F2, 1) (dual of [16, 15, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(21, s, F2, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(32) ⊂ Ce(30) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(2242, 32784, F2, 33) (dual of [32784, 32542, 34]-code), using
- OOA 16-folding and stacking with additional row [i] based on linear OA(2243, 32785, F2, 33) (dual of [32785, 32542, 34]-code), using
- net defined by OOA [i] based on linear OOA(2243, 2049, F2, 33, 33) (dual of [(2049, 33), 67374, 34]-NRT-code), using
(244−33, 244, 5464)-Net over F2 — Digital
Digital (211, 244, 5464)-net over F2, using
- 22 times duplication [i] based on digital (209, 242, 5464)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2242, 5464, F2, 6, 33) (dual of [(5464, 6), 32542, 34]-NRT-code), using
- OOA 6-folding [i] based on linear OA(2242, 32784, F2, 33) (dual of [32784, 32542, 34]-code), using
- construction X applied to Ce(32) ⊂ Ce(30) [i] based on
- linear OA(2241, 32768, F2, 33) (dual of [32768, 32527, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(2226, 32768, F2, 31) (dual of [32768, 32542, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(21, 16, F2, 1) (dual of [16, 15, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(21, s, F2, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(32) ⊂ Ce(30) [i] based on
- OOA 6-folding [i] based on linear OA(2242, 32784, F2, 33) (dual of [32784, 32542, 34]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2242, 5464, F2, 6, 33) (dual of [(5464, 6), 32542, 34]-NRT-code), using
(244−33, 244, 253741)-Net in Base 2 — Upper bound on s
There is no (211, 244, 253742)-net in base 2, because
- 1 times m-reduction [i] would yield (211, 243, 253742)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 14 135189 472446 530932 890529 671267 405330 448312 249991 847272 585995 314160 795114 > 2243 [i]