Best Known (126−19, 126, s)-Nets in Base 2
(126−19, 126, 913)-Net over F2 — Constructive and digital
Digital (107, 126, 913)-net over F2, using
- 22 times duplication [i] based on digital (105, 124, 913)-net over F2, using
- net defined by OOA [i] based on linear OOA(2124, 913, F2, 19, 19) (dual of [(913, 19), 17223, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(2124, 8218, F2, 19) (dual of [8218, 8094, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(2124, 8224, F2, 19) (dual of [8224, 8100, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(14) [i] based on
- linear OA(2118, 8192, F2, 19) (dual of [8192, 8074, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(292, 8192, F2, 15) (dual of [8192, 8100, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(26, 32, F2, 3) (dual of [32, 26, 4]-code or 32-cap in PG(5,2)), using
- construction X applied to Ce(18) ⊂ Ce(14) [i] based on
- discarding factors / shortening the dual code based on linear OA(2124, 8224, F2, 19) (dual of [8224, 8100, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(2124, 8218, F2, 19) (dual of [8218, 8094, 20]-code), using
- net defined by OOA [i] based on linear OOA(2124, 913, F2, 19, 19) (dual of [(913, 19), 17223, 20]-NRT-code), using
(126−19, 126, 2056)-Net over F2 — Digital
Digital (107, 126, 2056)-net over F2, using
- 22 times duplication [i] based on digital (105, 124, 2056)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2124, 2056, F2, 4, 19) (dual of [(2056, 4), 8100, 20]-NRT-code), using
- OOA 4-folding [i] based on linear OA(2124, 8224, F2, 19) (dual of [8224, 8100, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(14) [i] based on
- linear OA(2118, 8192, F2, 19) (dual of [8192, 8074, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(292, 8192, F2, 15) (dual of [8192, 8100, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(26, 32, F2, 3) (dual of [32, 26, 4]-code or 32-cap in PG(5,2)), using
- construction X applied to Ce(18) ⊂ Ce(14) [i] based on
- OOA 4-folding [i] based on linear OA(2124, 8224, F2, 19) (dual of [8224, 8100, 20]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2124, 2056, F2, 4, 19) (dual of [(2056, 4), 8100, 20]-NRT-code), using
(126−19, 126, 62897)-Net in Base 2 — Upper bound on s
There is no (107, 126, 62898)-net in base 2, because
- 1 times m-reduction [i] would yield (107, 125, 62898)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 42 537914 082144 724075 097564 050742 126214 > 2125 [i]