Best Known (109−9, 109, s)-Nets in Base 7
(109−9, 109, large)-Net over F7 — Constructive and digital
Digital (100, 109, large)-net over F7, using
- 72 times duplication [i] based on digital (98, 107, large)-net over F7, using
- generalized (u, u+v)-construction [i] based on
- digital (0, 1, 1441202)-net over F7, using
- s-reduction based on digital (0, 1, s)-net over F7 with arbitrarily large s, using
- digital (0, 1, 1441202)-net over F7 (see above)
- digital (0, 1, 1441202)-net over F7 (see above)
- digital (7, 9, 1441202)-net over F7, using
- s-reduction based on digital (7, 9, 6725601)-net over F7, using
- digital (10, 13, 1441202)-net over F7, using
- s-reduction based on digital (10, 13, 6120101)-net over F7, using
- net defined by OOA [i] based on linear OOA(713, 6120101, F7, 3, 3) (dual of [(6120101, 3), 18360290, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(713, 6120101, F7, 2, 3) (dual of [(6120101, 2), 12240189, 4]-NRT-code), using
- OAs with strength 3, b ≠ 2, and m > 3 are always embeddable [i] based on linear OA(713, 6120101, F7, 3) (dual of [6120101, 6120088, 4]-code or 6120101-cap in PG(12,7)), using
- appending kth column [i] based on linear OOA(713, 6120101, F7, 2, 3) (dual of [(6120101, 2), 12240189, 4]-NRT-code), using
- net defined by OOA [i] based on linear OOA(713, 6120101, F7, 3, 3) (dual of [(6120101, 3), 18360290, 4]-NRT-code), using
- s-reduction based on digital (10, 13, 6120101)-net over F7, using
- digital (21, 25, 1441202)-net over F7, using
- s-reduction based on digital (21, 25, 2882404)-net over F7, using
- net defined by OOA [i] based on linear OOA(725, 2882404, F7, 4, 4) (dual of [(2882404, 4), 11529591, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(725, 2882404, F7, 3, 4) (dual of [(2882404, 3), 8647187, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(725, 5764808, F7, 4) (dual of [5764808, 5764783, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(725, 5764809, F7, 4) (dual of [5764809, 5764784, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- linear OA(725, 5764801, F7, 4) (dual of [5764801, 5764776, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 78−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(717, 5764801, F7, 3) (dual of [5764801, 5764784, 4]-code or 5764801-cap in PG(16,7)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 78−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(70, 8, F7, 0) (dual of [8, 8, 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(3) ⊂ Ce(2) [i] based on
- discarding factors / shortening the dual code based on linear OA(725, 5764809, F7, 4) (dual of [5764809, 5764784, 5]-code), using
- OA 2-folding and stacking [i] based on linear OA(725, 5764808, F7, 4) (dual of [5764808, 5764783, 5]-code), using
- appending kth column [i] based on linear OOA(725, 2882404, F7, 3, 4) (dual of [(2882404, 3), 8647187, 5]-NRT-code), using
- net defined by OOA [i] based on linear OOA(725, 2882404, F7, 4, 4) (dual of [(2882404, 4), 11529591, 5]-NRT-code), using
- s-reduction based on digital (21, 25, 2882404)-net over F7, using
- digital (48, 57, 1441202)-net over F7, using
- net defined by OOA [i] based on linear OOA(757, 1441202, F7, 9, 9) (dual of [(1441202, 9), 12970761, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(757, 5764809, F7, 9) (dual of [5764809, 5764752, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(7) [i] based on
- linear OA(757, 5764801, F7, 9) (dual of [5764801, 5764744, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 78−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(749, 5764801, F7, 8) (dual of [5764801, 5764752, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 78−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(70, 8, F7, 0) (dual of [8, 8, 1]-code) (see above)
- construction X applied to Ce(8) ⊂ Ce(7) [i] based on
- OOA 4-folding and stacking with additional row [i] based on linear OA(757, 5764809, F7, 9) (dual of [5764809, 5764752, 10]-code), using
- net defined by OOA [i] based on linear OOA(757, 1441202, F7, 9, 9) (dual of [(1441202, 9), 12970761, 10]-NRT-code), using
- digital (0, 1, 1441202)-net over F7, using
- generalized (u, u+v)-construction [i] based on
(109−9, 109, large)-Net in Base 7 — Upper bound on s
There is no (100, 109, large)-net in base 7, because
- 7 times m-reduction [i] would yield (100, 102, large)-net in base 7, but