Best Known (69, 69+15, s)-Nets in Base 2
(69, 69+15, 296)-Net over F2 — Constructive and digital
Digital (69, 84, 296)-net over F2, using
- net defined by OOA [i] based on linear OOA(284, 296, F2, 15, 15) (dual of [(296, 15), 4356, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(284, 2073, F2, 15) (dual of [2073, 1989, 16]-code), using
- discarding factors / shortening the dual code 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
- discarding factors / shortening the dual code based on linear OA(284, 2076, F2, 15) (dual of [2076, 1992, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(284, 2073, F2, 15) (dual of [2073, 1989, 16]-code), using
(69, 69+15, 692)-Net over F2 — Digital
Digital (69, 84, 692)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(284, 692, F2, 3, 15) (dual of [(692, 3), 1992, 16]-NRT-code), using
- OOA 3-folding [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
- OOA 3-folding [i] based on linear OA(284, 2076, F2, 15) (dual of [2076, 1992, 16]-code), using
(69, 69+15, 12529)-Net in Base 2 — Upper bound on s
There is no (69, 84, 12530)-net in base 2, because
- 1 times m-reduction [i] would yield (69, 83, 12530)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 9 675008 800089 951741 158952 > 283 [i]