Best Known (67, 78, s)-Nets in Base 2
(67, 78, 6557)-Net over F2 — Constructive and digital
Digital (67, 78, 6557)-net over F2, using
- net defined by OOA [i] based on linear OOA(278, 6557, F2, 11, 11) (dual of [(6557, 11), 72049, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(278, 32786, F2, 11) (dual of [32786, 32708, 12]-code), using
- 1 times code embedding in larger space [i] based on linear OA(277, 32785, F2, 11) (dual of [32785, 32708, 12]-code), using
- construction X4 applied to Ce(10) ⊂ Ce(8) [i] based on
- linear OA(276, 32768, F2, 11) (dual of [32768, 32692, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(261, 32768, F2, 9) (dual of [32768, 32707, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [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(10) ⊂ Ce(8) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(277, 32785, F2, 11) (dual of [32785, 32708, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(278, 32786, F2, 11) (dual of [32786, 32708, 12]-code), using
(67, 78, 8196)-Net over F2 — Digital
Digital (67, 78, 8196)-net over F2, using
- 21 times duplication [i] based on digital (66, 77, 8196)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(277, 8196, F2, 4, 11) (dual of [(8196, 4), 32707, 12]-NRT-code), using
- OOA 4-folding [i] based on linear OA(277, 32784, F2, 11) (dual of [32784, 32707, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(8) [i] based on
- linear OA(276, 32768, F2, 11) (dual of [32768, 32692, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(261, 32768, F2, 9) (dual of [32768, 32707, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [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(10) ⊂ Ce(8) [i] based on
- OOA 4-folding [i] based on linear OA(277, 32784, F2, 11) (dual of [32784, 32707, 12]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(277, 8196, F2, 4, 11) (dual of [(8196, 4), 32707, 12]-NRT-code), using
(67, 78, 112634)-Net in Base 2 — Upper bound on s
There is no (67, 78, 112635)-net in base 2, because
- 1 times m-reduction [i] would yield (67, 77, 112635)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 151119 502431 395207 446528 > 277 [i]