Best Known (89, 89+8, s)-Nets in Base 7
(89, 89+8, large)-Net over F7 — Constructive and digital
Digital (89, 97, large)-net over F7, using
- 72 times duplication [i] based on digital (87, 95, large)-net over F7, using
- generalized (u, u+v)-construction [i] based on
- digital (0, 1, 1441200)-net over F7, using
- s-reduction based on digital (0, 1, s)-net over F7 with arbitrarily large s, using
- digital (0, 1, 1441200)-net over F7 (see above)
- digital (0, 1, 1441200)-net over F7 (see above)
- digital (7, 9, 1441200)-net over F7, using
- s-reduction based on digital (7, 9, 6725601)-net over F7, using
- digital (7, 9, 1441200)-net over F7 (see above)
- digital (21, 25, 1441200)-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 (41, 49, 1441200)-net over F7, using
- net defined by OOA [i] based on linear OOA(749, 1441200, F7, 8, 8) (dual of [(1441200, 8), 11529551, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(749, 5764800, F7, 8) (dual of [5764800, 5764751, 9]-code), using
- discarding factors / shortening the dual code based on 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]
- discarding factors / shortening the dual code based on linear OA(749, 5764801, F7, 8) (dual of [5764801, 5764752, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(749, 5764800, F7, 8) (dual of [5764800, 5764751, 9]-code), using
- net defined by OOA [i] based on linear OOA(749, 1441200, F7, 8, 8) (dual of [(1441200, 8), 11529551, 9]-NRT-code), using
- digital (0, 1, 1441200)-net over F7, using
- generalized (u, u+v)-construction [i] based on
(89, 89+8, large)-Net in Base 7 — Upper bound on s
There is no (89, 97, large)-net in base 7, because
- 6 times m-reduction [i] would yield (89, 91, large)-net in base 7, but