Best Known (70−7, 70, s)-Nets in Base 7
(70−7, 70, 7823661)-Net over F7 — Constructive and digital
Digital (63, 70, 7823661)-net over F7, using
- generalized (u, u+v)-construction [i] based on
- digital (5, 7, 137257)-net over F7, using
- digital (10, 13, 3843202)-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 (43, 50, 3843202)-net over F7, using
- trace code for nets [i] based on digital (18, 25, 1921601)-net over F49, using
- net defined by OOA [i] based on linear OOA(4925, 1921601, F49, 7, 7) (dual of [(1921601, 7), 13451182, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(4925, 5764804, F49, 7) (dual of [5764804, 5764779, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(4925, 5764805, F49, 7) (dual of [5764805, 5764780, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- 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(4921, 5764801, F49, 6) (dual of [5764801, 5764780, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 494−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(490, 4, F49, 0) (dual of [4, 4, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(490, s, F49, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- discarding factors / shortening the dual code based on linear OA(4925, 5764805, F49, 7) (dual of [5764805, 5764780, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(4925, 5764804, F49, 7) (dual of [5764804, 5764779, 8]-code), using
- net defined by OOA [i] based on linear OOA(4925, 1921601, F49, 7, 7) (dual of [(1921601, 7), 13451182, 8]-NRT-code), using
- trace code for nets [i] based on digital (18, 25, 1921601)-net over F49, using
(70−7, 70, large)-Net over F7 — Digital
Digital (63, 70, large)-net over F7, using
- 3 times m-reduction [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
(70−7, 70, large)-Net in Base 7 — Upper bound on s
There is no (63, 70, large)-net in base 7, because
- 5 times m-reduction [i] would yield (63, 65, large)-net in base 7, but