Best Known (68, 78, s)-Nets in Base 7
(68, 78, 2305926)-Net over F7 — Constructive and digital
Digital (68, 78, 2305926)-net over F7, using
- trace code for nets [i] based on digital (29, 39, 1152963)-net over F49, using
- net defined by OOA [i] based on linear OOA(4939, 1152963, F49, 10, 10) (dual of [(1152963, 10), 11529591, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(4939, 5764815, F49, 10) (dual of [5764815, 5764776, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(6) [i] based on
- linear OA(4937, 5764801, F49, 10) (dual of [5764801, 5764764, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 494−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(4925, 5764801, F49, 7) (dual of [5764801, 5764776, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 494−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(492, 14, F49, 2) (dual of [14, 12, 3]-code or 14-arc in PG(1,49)), using
- discarding factors / shortening the dual code based on linear OA(492, 49, F49, 2) (dual of [49, 47, 3]-code or 49-arc in PG(1,49)), using
- Reed–Solomon code RS(47,49) [i]
- discarding factors / shortening the dual code based on linear OA(492, 49, F49, 2) (dual of [49, 47, 3]-code or 49-arc in PG(1,49)), using
- construction X applied to Ce(9) ⊂ Ce(6) [i] based on
- OA 5-folding and stacking [i] based on linear OA(4939, 5764815, F49, 10) (dual of [5764815, 5764776, 11]-code), using
- net defined by OOA [i] based on linear OOA(4939, 1152963, F49, 10, 10) (dual of [(1152963, 10), 11529591, 11]-NRT-code), using
(68, 78, large)-Net over F7 — Digital
Digital (68, 78, large)-net over F7, using
- 75 times duplication [i] based on digital (63, 73, large)-net over F7, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(773, large, F7, 10) (dual of [large, large−73, 11]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 20176803 | 79−1, defining interval I = [0,9], and designed minimum distance d ≥ |I|+1 = 11 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(773, large, F7, 10) (dual of [large, large−73, 11]-code), using
(68, 78, large)-Net in Base 7 — Upper bound on s
There is no (68, 78, large)-net in base 7, because
- 8 times m-reduction [i] would yield (68, 70, large)-net in base 7, but