Best Known (72, 72+13, s)-Nets in Base 7
(72, 72+13, 137271)-Net over F7 — Constructive and digital
Digital (72, 85, 137271)-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 (65, 78, 137258)-net over F7, using
- net defined by OOA [i] based on linear OOA(778, 137258, F7, 13, 13) (dual of [(137258, 13), 1784276, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(778, 823549, F7, 13) (dual of [823549, 823471, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(778, 823550, F7, 13) (dual of [823550, 823472, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- linear OA(778, 823543, F7, 13) (dual of [823543, 823465, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- 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(70, 7, F7, 0) (dual of [7, 7, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(70, s, F7, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(778, 823550, F7, 13) (dual of [823550, 823472, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(778, 823549, F7, 13) (dual of [823549, 823471, 14]-code), using
- net defined by OOA [i] based on linear OOA(778, 137258, F7, 13, 13) (dual of [(137258, 13), 1784276, 14]-NRT-code), using
- digital (1, 7, 13)-net over F7, using
(72, 72+13, 853737)-Net over F7 — Digital
Digital (72, 85, 853737)-net over F7, using
(72, 72+13, large)-Net in Base 7 — Upper bound on s
There is no (72, 85, large)-net in base 7, because
- 11 times m-reduction [i] would yield (72, 74, large)-net in base 7, but