Best Known (74−12, 74, s)-Nets in Base 7
(74−12, 74, 137261)-Net over F7 — Constructive and digital
Digital (62, 74, 137261)-net over F7, using
- net defined by OOA [i] based on linear OOA(774, 137261, F7, 12, 12) (dual of [(137261, 12), 1647058, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(774, 823566, F7, 12) (dual of [823566, 823492, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(774, 823567, F7, 12) (dual of [823567, 823493, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(8) [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(750, 823543, F7, 9) (dual of [823543, 823493, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(73, 24, F7, 2) (dual of [24, 21, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(73, 48, F7, 2) (dual of [48, 45, 3]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 48 = 72−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- discarding factors / shortening the dual code based on linear OA(73, 48, F7, 2) (dual of [48, 45, 3]-code), using
- construction X applied to Ce(11) ⊂ Ce(8) [i] based on
- discarding factors / shortening the dual code based on linear OA(774, 823567, F7, 12) (dual of [823567, 823493, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(774, 823566, F7, 12) (dual of [823566, 823492, 13]-code), using
(74−12, 74, 823567)-Net over F7 — Digital
Digital (62, 74, 823567)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(774, 823567, F7, 12) (dual of [823567, 823493, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(8) [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(750, 823543, F7, 9) (dual of [823543, 823493, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(73, 24, F7, 2) (dual of [24, 21, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(73, 48, F7, 2) (dual of [48, 45, 3]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 48 = 72−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- discarding factors / shortening the dual code based on linear OA(73, 48, F7, 2) (dual of [48, 45, 3]-code), using
- construction X applied to Ce(11) ⊂ Ce(8) [i] based on
(74−12, 74, large)-Net in Base 7 — Upper bound on s
There is no (62, 74, large)-net in base 7, because
- 10 times m-reduction [i] would yield (62, 64, large)-net in base 7, but