Best Known (67, 67+8, s)-Nets in Base 2
(67, 67+8, 65541)-Net over F2 — Constructive and digital
Digital (67, 75, 65541)-net over F2, using
- t-expansion [i] based on digital (66, 75, 65541)-net over F2, using
- net defined by OOA [i] based on linear OOA(275, 65541, F2, 9, 9) (dual of [(65541, 9), 589794, 10]-NRT-code), using
- appending kth column [i] based on linear OOA(275, 65541, F2, 8, 9) (dual of [(65541, 8), 524253, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(275, 262165, F2, 9) (dual of [262165, 262090, 10]-code), using
- 1 times code embedding in larger space [i] based on linear OA(274, 262164, F2, 9) (dual of [262164, 262090, 10]-code), using
- construction X4 applied to Ce(8) ⊂ Ce(6) [i] based on
- linear OA(273, 262144, F2, 9) (dual of [262144, 262071, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 262143 = 218−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(255, 262144, F2, 7) (dual of [262144, 262089, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 262143 = 218−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(219, 20, F2, 19) (dual of [20, 1, 20]-code or 20-arc in PG(18,2)), using
- dual of repetition code with length 20 [i]
- linear OA(21, 20, F2, 1) (dual of [20, 19, 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(8) ⊂ Ce(6) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(274, 262164, F2, 9) (dual of [262164, 262090, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(275, 262165, F2, 9) (dual of [262165, 262090, 10]-code), using
- appending kth column [i] based on linear OOA(275, 65541, F2, 8, 9) (dual of [(65541, 8), 524253, 10]-NRT-code), using
- net defined by OOA [i] based on linear OOA(275, 65541, F2, 9, 9) (dual of [(65541, 9), 589794, 10]-NRT-code), using
(67, 67+8, 87388)-Net over F2 — Digital
Digital (67, 75, 87388)-net over F2, using
- 21 times duplication [i] based on digital (66, 74, 87388)-net over F2, using
- net defined by OOA [i] based on linear OOA(274, 87388, F2, 8, 8) (dual of [(87388, 8), 699030, 9]-NRT-code), using
- appending kth column [i] based on linear OOA(274, 87388, F2, 7, 8) (dual of [(87388, 7), 611642, 9]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(274, 87388, F2, 3, 8) (dual of [(87388, 3), 262090, 9]-NRT-code), using
- OOA 3-folding [i] based on linear OA(274, 262164, F2, 8) (dual of [262164, 262090, 9]-code), using
- strength reduction [i] based on linear OA(274, 262164, F2, 9) (dual of [262164, 262090, 10]-code), using
- construction X4 applied to Ce(8) ⊂ Ce(6) [i] based on
- linear OA(273, 262144, F2, 9) (dual of [262144, 262071, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 262143 = 218−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(255, 262144, F2, 7) (dual of [262144, 262089, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 262143 = 218−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(219, 20, F2, 19) (dual of [20, 1, 20]-code or 20-arc in PG(18,2)), using
- dual of repetition code with length 20 [i]
- linear OA(21, 20, F2, 1) (dual of [20, 19, 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(8) ⊂ Ce(6) [i] based on
- strength reduction [i] based on linear OA(274, 262164, F2, 9) (dual of [262164, 262090, 10]-code), using
- OOA 3-folding [i] based on linear OA(274, 262164, F2, 8) (dual of [262164, 262090, 9]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(274, 87388, F2, 3, 8) (dual of [(87388, 3), 262090, 9]-NRT-code), using
- appending kth column [i] based on linear OOA(274, 87388, F2, 7, 8) (dual of [(87388, 7), 611642, 9]-NRT-code), using
- net defined by OOA [i] based on linear OOA(274, 87388, F2, 8, 8) (dual of [(87388, 8), 699030, 9]-NRT-code), using
(67, 67+8, 975804)-Net in Base 2 — Upper bound on s
There is no (67, 75, 975805)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 37779 021566 641421 138771 > 275 [i]