Best Known (168, 168+23, s)-Nets in Base 2
(168, 168+23, 11917)-Net over F2 — Constructive and digital
Digital (168, 191, 11917)-net over F2, using
- 22 times duplication [i] based on digital (166, 189, 11917)-net over F2, using
- net defined by OOA [i] based on linear OOA(2189, 11917, F2, 23, 23) (dual of [(11917, 23), 273902, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(2189, 131088, F2, 23) (dual of [131088, 130899, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(2189, 131090, F2, 23) (dual of [131090, 130901, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(20) [i] based on
- linear OA(2188, 131072, F2, 23) (dual of [131072, 130884, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 131071 = 217−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(2171, 131072, F2, 21) (dual of [131072, 130901, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 131071 = 217−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(21, 18, F2, 1) (dual of [18, 17, 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(2189, 131090, F2, 23) (dual of [131090, 130901, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(2189, 131088, F2, 23) (dual of [131088, 130899, 24]-code), using
- net defined by OOA [i] based on linear OOA(2189, 11917, F2, 23, 23) (dual of [(11917, 23), 273902, 24]-NRT-code), using
(168, 168+23, 20545)-Net over F2 — Digital
Digital (168, 191, 20545)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2191, 20545, F2, 6, 23) (dual of [(20545, 6), 123079, 24]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2191, 21848, F2, 6, 23) (dual of [(21848, 6), 130897, 24]-NRT-code), using
- 22 times duplication [i] based on linear OOA(2189, 21848, F2, 6, 23) (dual of [(21848, 6), 130899, 24]-NRT-code), using
- OOA 6-folding [i] based on linear OA(2189, 131088, F2, 23) (dual of [131088, 130899, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(2189, 131090, F2, 23) (dual of [131090, 130901, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(20) [i] based on
- linear OA(2188, 131072, F2, 23) (dual of [131072, 130884, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 131071 = 217−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(2171, 131072, F2, 21) (dual of [131072, 130901, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 131071 = 217−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(21, 18, F2, 1) (dual of [18, 17, 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(2189, 131090, F2, 23) (dual of [131090, 130901, 24]-code), using
- OOA 6-folding [i] based on linear OA(2189, 131088, F2, 23) (dual of [131088, 130899, 24]-code), using
- 22 times duplication [i] based on linear OOA(2189, 21848, F2, 6, 23) (dual of [(21848, 6), 130899, 24]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2191, 21848, F2, 6, 23) (dual of [(21848, 6), 130897, 24]-NRT-code), using
(168, 168+23, 777345)-Net in Base 2 — Upper bound on s
There is no (168, 191, 777346)-net in base 2, because
- 1 times m-reduction [i] would yield (168, 190, 777346)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 1569 280791 627683 657397 057105 627105 959398 364295 859926 210432 > 2190 [i]