Best Known (109−15, 109, s)-Nets in Base 2
(109−15, 109, 4683)-Net over F2 — Constructive and digital
Digital (94, 109, 4683)-net over F2, using
- 22 times duplication [i] based on digital (92, 107, 4683)-net over F2, using
- net defined by OOA [i] based on linear OOA(2107, 4683, F2, 15, 15) (dual of [(4683, 15), 70138, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(2107, 32782, F2, 15) (dual of [32782, 32675, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(2107, 32784, F2, 15) (dual of [32784, 32677, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(12) [i] based on
- linear OA(2106, 32768, F2, 15) (dual of [32768, 32662, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(291, 32768, F2, 13) (dual of [32768, 32677, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [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(14) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(2107, 32784, F2, 15) (dual of [32784, 32677, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(2107, 32782, F2, 15) (dual of [32782, 32675, 16]-code), using
- net defined by OOA [i] based on linear OOA(2107, 4683, F2, 15, 15) (dual of [(4683, 15), 70138, 16]-NRT-code), using
(109−15, 109, 6557)-Net over F2 — Digital
Digital (94, 109, 6557)-net over F2, using
- 22 times duplication [i] based on digital (92, 107, 6557)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2107, 6557, F2, 5, 15) (dual of [(6557, 5), 32678, 16]-NRT-code), using
- OOA 5-folding [i] based on linear OA(2107, 32785, F2, 15) (dual of [32785, 32678, 16]-code), using
- construction X4 applied to Ce(14) ⊂ Ce(12) [i] based on
- linear OA(2106, 32768, F2, 15) (dual of [32768, 32662, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(291, 32768, F2, 13) (dual of [32768, 32677, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(216, 17, F2, 15) (dual of [17, 1, 16]-code), using
- strength reduction [i] based on linear OA(216, 17, F2, 16) (dual of [17, 1, 17]-code or 17-arc in PG(15,2)), using
- dual of repetition code with length 17 [i]
- strength reduction [i] based on linear OA(216, 17, F2, 16) (dual of [17, 1, 17]-code or 17-arc in PG(15,2)), using
- linear OA(21, 17, F2, 1) (dual of [17, 16, 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 X4 applied to Ce(14) ⊂ Ce(12) [i] based on
- OOA 5-folding [i] based on linear OA(2107, 32785, F2, 15) (dual of [32785, 32678, 16]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2107, 6557, F2, 5, 15) (dual of [(6557, 5), 32678, 16]-NRT-code), using
(109−15, 109, 149056)-Net in Base 2 — Upper bound on s
There is no (94, 109, 149057)-net in base 2, because
- 1 times m-reduction [i] would yield (94, 108, 149057)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 324 518930 950008 081010 478199 394944 > 2108 [i]