Best Known (191−21, 191, s)-Nets in Base 2
(191−21, 191, 52428)-Net over F2 — Constructive and digital
Digital (170, 191, 52428)-net over F2, using
- net defined by OOA [i] based on linear OOA(2191, 52428, F2, 21, 21) (dual of [(52428, 21), 1100797, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(2191, 524281, F2, 21) (dual of [524281, 524090, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(2191, 524287, F2, 21) (dual of [524287, 524096, 22]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 524287 = 219−1, defining interval I = [0,20], and designed minimum distance d ≥ |I|+1 = 22 [i]
- discarding factors / shortening the dual code based on linear OA(2191, 524287, F2, 21) (dual of [524287, 524096, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(2191, 524281, F2, 21) (dual of [524281, 524090, 22]-code), using
(191−21, 191, 74898)-Net over F2 — Digital
Digital (170, 191, 74898)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2191, 74898, F2, 7, 21) (dual of [(74898, 7), 524095, 22]-NRT-code), using
- OOA 7-folding [i] based on linear OA(2191, 524286, F2, 21) (dual of [524286, 524095, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(2191, 524287, F2, 21) (dual of [524287, 524096, 22]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 524287 = 219−1, defining interval I = [0,20], and designed minimum distance d ≥ |I|+1 = 22 [i]
- discarding factors / shortening the dual code based on linear OA(2191, 524287, F2, 21) (dual of [524287, 524096, 22]-code), using
- OOA 7-folding [i] based on linear OA(2191, 524286, F2, 21) (dual of [524286, 524095, 22]-code), using
(191−21, 191, 2374343)-Net in Base 2 — Upper bound on s
There is no (170, 191, 2374344)-net in base 2, because
- 1 times m-reduction [i] would yield (170, 190, 2374344)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 1569 278034 758330 985829 646854 282283 141632 831172 585919 472344 > 2190 [i]