Best Known (214, 247, s)-Nets in Base 2
(214, 247, 2050)-Net over F2 — Constructive and digital
Digital (214, 247, 2050)-net over F2, using
- net defined by OOA [i] based on linear OOA(2247, 2050, F2, 33, 33) (dual of [(2050, 33), 67403, 34]-NRT-code), using
- OOA 16-folding and stacking with additional row [i] based on linear OA(2247, 32801, F2, 33) (dual of [32801, 32554, 34]-code), using
- construction X applied to C([0,16]) ⊂ C([0,14]) [i] based on
- linear OA(2241, 32769, F2, 33) (dual of [32769, 32528, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 230−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- linear OA(2211, 32769, F2, 29) (dual of [32769, 32558, 30]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 230−1, defining interval I = [0,14], and minimum distance d ≥ |{−14,−13,…,14}|+1 = 30 (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([0,16]) ⊂ C([0,14]) [i] based on
- OOA 16-folding and stacking with additional row [i] based on linear OA(2247, 32801, F2, 33) (dual of [32801, 32554, 34]-code), using
(214, 247, 5466)-Net over F2 — Digital
Digital (214, 247, 5466)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2247, 5466, F2, 6, 33) (dual of [(5466, 6), 32549, 34]-NRT-code), using
- OOA 6-folding [i] based on linear OA(2247, 32796, F2, 33) (dual of [32796, 32549, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(2247, 32801, F2, 33) (dual of [32801, 32554, 34]-code), using
- construction X applied to C([0,16]) ⊂ C([0,14]) [i] based on
- linear OA(2241, 32769, F2, 33) (dual of [32769, 32528, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 230−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- linear OA(2211, 32769, F2, 29) (dual of [32769, 32558, 30]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 230−1, defining interval I = [0,14], and minimum distance d ≥ |{−14,−13,…,14}|+1 = 30 (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([0,16]) ⊂ C([0,14]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2247, 32801, F2, 33) (dual of [32801, 32554, 34]-code), using
- OOA 6-folding [i] based on linear OA(2247, 32796, F2, 33) (dual of [32796, 32549, 34]-code), using
(214, 247, 288961)-Net in Base 2 — Upper bound on s
There is no (214, 247, 288962)-net in base 2, because
- 1 times m-reduction [i] would yield (214, 246, 288962)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 113 082974 350227 848413 913099 203587 869071 716475 660898 176584 399499 133892 639444 > 2246 [i]