Best Known (67, 67+10, s)-Nets in Base 7
(67, 67+10, 2305924)-Net over F7 — Constructive and digital
Digital (67, 77, 2305924)-net over F7, using
- 71 times duplication [i] based on digital (66, 76, 2305924)-net over F7, using
- trace code for nets [i] based on digital (28, 38, 1152962)-net over F49, using
- net defined by OOA [i] based on linear OOA(4938, 1152962, F49, 10, 10) (dual of [(1152962, 10), 11529582, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(4938, 5764810, F49, 10) (dual of [5764810, 5764772, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(7) [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(4929, 5764801, F49, 8) (dual of [5764801, 5764772, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 494−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(491, 9, F49, 1) (dual of [9, 8, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(491, s, F49, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(9) ⊂ Ce(7) [i] based on
- OA 5-folding and stacking [i] based on linear OA(4938, 5764810, F49, 10) (dual of [5764810, 5764772, 11]-code), using
- net defined by OOA [i] based on linear OOA(4938, 1152962, F49, 10, 10) (dual of [(1152962, 10), 11529582, 11]-NRT-code), using
- trace code for nets [i] based on digital (28, 38, 1152962)-net over F49, using
(67, 67+10, large)-Net over F7 — Digital
Digital (67, 77, large)-net over F7, using
- 74 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
(67, 67+10, large)-Net in Base 7 — Upper bound on s
There is no (67, 77, large)-net in base 7, because
- 8 times m-reduction [i] would yield (67, 69, large)-net in base 7, but