Best Known (168−22, 168, s)-Nets in Base 2
(168−22, 168, 2980)-Net over F2 — Constructive and digital
Digital (146, 168, 2980)-net over F2, using
- 21 times duplication [i] based on digital (145, 167, 2980)-net over F2, using
- t-expansion [i] based on digital (144, 167, 2980)-net over F2, using
- net defined by OOA [i] based on linear OOA(2167, 2980, F2, 23, 23) (dual of [(2980, 23), 68373, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(2167, 32781, F2, 23) (dual of [32781, 32614, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(2167, 32784, F2, 23) (dual of [32784, 32617, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(20) [i] based on
- linear OA(2166, 32768, F2, 23) (dual of [32768, 32602, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(2151, 32768, F2, 21) (dual of [32768, 32617, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [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(22) ⊂ Ce(20) [i] based on
- discarding factors / shortening the dual code based on linear OA(2167, 32784, F2, 23) (dual of [32784, 32617, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(2167, 32781, F2, 23) (dual of [32781, 32614, 24]-code), using
- net defined by OOA [i] based on linear OOA(2167, 2980, F2, 23, 23) (dual of [(2980, 23), 68373, 24]-NRT-code), using
- t-expansion [i] based on digital (144, 167, 2980)-net over F2, using
(168−22, 168, 6557)-Net over F2 — Digital
Digital (146, 168, 6557)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2168, 6557, F2, 5, 22) (dual of [(6557, 5), 32617, 23]-NRT-code), using
- OOA 5-folding [i] based on linear OA(2168, 32785, F2, 22) (dual of [32785, 32617, 23]-code), using
- 2 times code embedding in larger space [i] based on linear OA(2166, 32783, F2, 22) (dual of [32783, 32617, 23]-code), using
- 1 times truncation [i] based on linear OA(2167, 32784, F2, 23) (dual of [32784, 32617, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(20) [i] based on
- linear OA(2166, 32768, F2, 23) (dual of [32768, 32602, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(2151, 32768, F2, 21) (dual of [32768, 32617, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [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(22) ⊂ Ce(20) [i] based on
- 1 times truncation [i] based on linear OA(2167, 32784, F2, 23) (dual of [32784, 32617, 24]-code), using
- 2 times code embedding in larger space [i] based on linear OA(2166, 32783, F2, 22) (dual of [32783, 32617, 23]-code), using
- OOA 5-folding [i] based on linear OA(2168, 32785, F2, 22) (dual of [32785, 32617, 23]-code), using
(168−22, 168, 194324)-Net in Base 2 — Upper bound on s
There is no (146, 168, 194325)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 374 156273 578072 862075 331616 229475 660085 872287 922176 > 2168 [i]