Best Known (210, 210+32, s)-Nets in Base 2
(210, 210+32, 2049)-Net over F2 — Constructive and digital
Digital (210, 242, 2049)-net over F2, using
- net defined by OOA [i] based on linear OOA(2242, 2049, F2, 32, 32) (dual of [(2049, 32), 65326, 33]-NRT-code), using
- OA 16-folding and stacking [i] based on linear OA(2242, 32784, F2, 32) (dual of [32784, 32542, 33]-code), using
- strength reduction [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
- strength reduction [i] based on linear OA(2242, 32784, F2, 33) (dual of [32784, 32542, 34]-code), using
- OA 16-folding and stacking [i] based on linear OA(2242, 32784, F2, 32) (dual of [32784, 32542, 33]-code), using
(210, 210+32, 5814)-Net over F2 — Digital
Digital (210, 242, 5814)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2242, 5814, F2, 5, 32) (dual of [(5814, 5), 28828, 33]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2242, 6556, F2, 5, 32) (dual of [(6556, 5), 32538, 33]-NRT-code), using
- strength reduction [i] based on linear OOA(2242, 6556, F2, 5, 33) (dual of [(6556, 5), 32538, 34]-NRT-code), using
- OOA 5-folding [i] based on linear OA(2242, 32780, F2, 33) (dual of [32780, 32538, 34]-code), using
- discarding factors / shortening the dual code 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
- discarding factors / shortening the dual code based on linear OA(2242, 32784, F2, 33) (dual of [32784, 32542, 34]-code), using
- OOA 5-folding [i] based on linear OA(2242, 32780, F2, 33) (dual of [32780, 32538, 34]-code), using
- strength reduction [i] based on linear OOA(2242, 6556, F2, 5, 33) (dual of [(6556, 5), 32538, 34]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2242, 6556, F2, 5, 32) (dual of [(6556, 5), 32538, 33]-NRT-code), using
(210, 210+32, 242982)-Net in Base 2 — Upper bound on s
There is no (210, 242, 242983)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 7 067513 109081 214584 415689 885145 265476 052041 407980 668174 471418 651776 099470 > 2242 [i]