Best Known (69, 83, s)-Nets in Base 2
(69, 83, 296)-Net over F2 — Constructive and digital
Digital (69, 83, 296)-net over F2, using
- net defined by OOA [i] based on linear OOA(283, 296, F2, 14, 14) (dual of [(296, 14), 4061, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(283, 2072, F2, 14) (dual of [2072, 1989, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(283, 2075, F2, 14) (dual of [2075, 1992, 15]-code), using
- 1 times truncation [i] based on linear OA(284, 2076, F2, 15) (dual of [2076, 1992, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(10) [i] based on
- linear OA(278, 2048, F2, 15) (dual of [2048, 1970, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(256, 2048, F2, 11) (dual of [2048, 1992, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(26, 28, F2, 3) (dual of [28, 22, 4]-code or 28-cap in PG(5,2)), using
- discarding factors / shortening the dual code based on 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(14) ⊂ Ce(10) [i] based on
- 1 times truncation [i] based on linear OA(284, 2076, F2, 15) (dual of [2076, 1992, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(283, 2075, F2, 14) (dual of [2075, 1992, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(283, 2072, F2, 14) (dual of [2072, 1989, 15]-code), using
(69, 83, 793)-Net over F2 — Digital
Digital (69, 83, 793)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(283, 793, F2, 2, 14) (dual of [(793, 2), 1503, 15]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(283, 1037, F2, 2, 14) (dual of [(1037, 2), 1991, 15]-NRT-code), using
- OOA 2-folding [i] based on linear OA(283, 2074, F2, 14) (dual of [2074, 1991, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(283, 2075, F2, 14) (dual of [2075, 1992, 15]-code), using
- 1 times truncation [i] based on linear OA(284, 2076, F2, 15) (dual of [2076, 1992, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(10) [i] based on
- linear OA(278, 2048, F2, 15) (dual of [2048, 1970, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(256, 2048, F2, 11) (dual of [2048, 1992, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(26, 28, F2, 3) (dual of [28, 22, 4]-code or 28-cap in PG(5,2)), using
- discarding factors / shortening the dual code based on 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(14) ⊂ Ce(10) [i] based on
- 1 times truncation [i] based on linear OA(284, 2076, F2, 15) (dual of [2076, 1992, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(283, 2075, F2, 14) (dual of [2075, 1992, 15]-code), using
- OOA 2-folding [i] based on linear OA(283, 2074, F2, 14) (dual of [2074, 1991, 15]-code), using
- discarding factors / shortening the dual code based on linear OOA(283, 1037, F2, 2, 14) (dual of [(1037, 2), 1991, 15]-NRT-code), using
(69, 83, 12529)-Net in Base 2 — Upper bound on s
There is no (69, 83, 12530)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 9 675008 800089 951741 158952 > 283 [i]