Best Known (247−25, 247, s)-Nets in Base 2
(247−25, 247, 87384)-Net over F2 — Constructive and digital
Digital (222, 247, 87384)-net over F2, using
- net defined by OOA [i] based on linear OOA(2247, 87384, F2, 25, 25) (dual of [(87384, 25), 2184353, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(2247, 1048609, F2, 25) (dual of [1048609, 1048362, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,10]) [i] based on
- linear OA(2241, 1048577, F2, 25) (dual of [1048577, 1048336, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 240−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(2201, 1048577, F2, 21) (dual of [1048577, 1048376, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 240−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [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 C([0,12]) ⊂ C([0,10]) [i] based on
- OOA 12-folding and stacking with additional row [i] based on linear OA(2247, 1048609, F2, 25) (dual of [1048609, 1048362, 26]-code), using
(247−25, 247, 131076)-Net over F2 — Digital
Digital (222, 247, 131076)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2247, 131076, F2, 8, 25) (dual of [(131076, 8), 1048361, 26]-NRT-code), using
- OOA 8-folding [i] based on linear OA(2247, 1048608, F2, 25) (dual of [1048608, 1048361, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(2247, 1048609, F2, 25) (dual of [1048609, 1048362, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,10]) [i] based on
- linear OA(2241, 1048577, F2, 25) (dual of [1048577, 1048336, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 240−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(2201, 1048577, F2, 21) (dual of [1048577, 1048376, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 240−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [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 C([0,12]) ⊂ C([0,10]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2247, 1048609, F2, 25) (dual of [1048609, 1048362, 26]-code), using
- OOA 8-folding [i] based on linear OA(2247, 1048608, F2, 25) (dual of [1048608, 1048361, 26]-code), using
(247−25, 247, 7842876)-Net in Base 2 — Upper bound on s
There is no (222, 247, 7842877)-net in base 2, because
- 1 times m-reduction [i] would yield (222, 246, 7842877)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 113 078362 991327 215735 889760 879569 288604 773169 063285 970987 287859 649548 457647 > 2246 [i]