Best Known (67, 67+12, s)-Nets in Base 7
(67, 67+12, 137272)-Net over F7 — Constructive and digital
Digital (67, 79, 137272)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (1, 7, 13)-net over F7, using
- 6 times m-reduction [i] based on digital (1, 13, 13)-net over F7, using
- digital (60, 72, 137259)-net over F7, using
- net defined by OOA [i] based on linear OOA(772, 137259, F7, 12, 12) (dual of [(137259, 12), 1647036, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(772, 823554, F7, 12) (dual of [823554, 823482, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(772, 823558, F7, 12) (dual of [823558, 823486, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(9) [i] based on
- linear OA(771, 823543, F7, 12) (dual of [823543, 823472, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(757, 823543, F7, 10) (dual of [823543, 823486, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(71, 15, F7, 1) (dual of [15, 14, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(71, s, F7, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(11) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(772, 823558, F7, 12) (dual of [823558, 823486, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(772, 823554, F7, 12) (dual of [823554, 823482, 13]-code), using
- net defined by OOA [i] based on linear OOA(772, 137259, F7, 12, 12) (dual of [(137259, 12), 1647036, 13]-NRT-code), using
- digital (1, 7, 13)-net over F7, using
(67, 67+12, 959855)-Net over F7 — Digital
Digital (67, 79, 959855)-net over F7, using
(67, 67+12, large)-Net in Base 7 — Upper bound on s
There is no (67, 79, large)-net in base 7, because
- 10 times m-reduction [i] would yield (67, 69, large)-net in base 7, but