Best Known (216−29, 216, s)-Nets in Base 2
(216−29, 216, 2341)-Net over F2 — Constructive and digital
Digital (187, 216, 2341)-net over F2, using
- 24 times duplication [i] based on digital (183, 212, 2341)-net over F2, using
- net defined by OOA [i] based on linear OOA(2212, 2341, F2, 29, 29) (dual of [(2341, 29), 67677, 30]-NRT-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(2212, 32775, F2, 29) (dual of [32775, 32563, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(2212, 32784, F2, 29) (dual of [32784, 32572, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(26) [i] based on
- linear OA(2211, 32768, F2, 29) (dual of [32768, 32557, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(2196, 32768, F2, 27) (dual of [32768, 32572, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(21, 16, F2, 1) (dual of [16, 15, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(21, s, F2, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(28) ⊂ Ce(26) [i] based on
- discarding factors / shortening the dual code based on linear OA(2212, 32784, F2, 29) (dual of [32784, 32572, 30]-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(2212, 32775, F2, 29) (dual of [32775, 32563, 30]-code), using
- net defined by OOA [i] based on linear OOA(2212, 2341, F2, 29, 29) (dual of [(2341, 29), 67677, 30]-NRT-code), using
(216−29, 216, 5464)-Net over F2 — Digital
Digital (187, 216, 5464)-net over F2, using
- 24 times duplication [i] based on digital (183, 212, 5464)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2212, 5464, F2, 6, 29) (dual of [(5464, 6), 32572, 30]-NRT-code), using
- OOA 6-folding [i] based on linear OA(2212, 32784, F2, 29) (dual of [32784, 32572, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(26) [i] based on
- linear OA(2211, 32768, F2, 29) (dual of [32768, 32557, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(2196, 32768, F2, 27) (dual of [32768, 32572, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(21, 16, F2, 1) (dual of [16, 15, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(21, s, F2, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(28) ⊂ Ce(26) [i] based on
- OOA 6-folding [i] based on linear OA(2212, 32784, F2, 29) (dual of [32784, 32572, 30]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2212, 5464, F2, 6, 29) (dual of [(5464, 6), 32572, 30]-NRT-code), using
(216−29, 216, 253736)-Net in Base 2 — Upper bound on s
There is no (187, 216, 253737)-net in base 2, because
- 1 times m-reduction [i] would yield (187, 215, 253737)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 52656 992197 062393 844366 393756 491006 250135 795712 597320 835448 968784 > 2215 [i]