Best Known (65−9, 65, s)-Nets in Base 7
(65−9, 65, 2097150)-Net over F7 — Constructive and digital
Digital (56, 65, 2097150)-net over F7, using
- 71 times duplication [i] based on digital (55, 64, 2097150)-net over F7, using
- net defined by OOA [i] based on linear OOA(764, 2097150, F7, 9, 9) (dual of [(2097150, 9), 18874286, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(764, 8388601, F7, 9) (dual of [8388601, 8388537, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(764, large, F7, 9) (dual of [large, large−64, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 20176803 | 79−1, defining interval I = [0,8], and designed minimum distance d ≥ |I|+1 = 10 [i]
- discarding factors / shortening the dual code based on linear OA(764, large, F7, 9) (dual of [large, large−64, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(764, 8388601, F7, 9) (dual of [8388601, 8388537, 10]-code), using
- net defined by OOA [i] based on linear OOA(764, 2097150, F7, 9, 9) (dual of [(2097150, 9), 18874286, 10]-NRT-code), using
(65−9, 65, large)-Net over F7 — Digital
Digital (56, 65, large)-net over F7, using
- 71 times duplication [i] based on digital (55, 64, large)-net over F7, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(764, large, F7, 9) (dual of [large, large−64, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 20176803 | 79−1, defining interval I = [0,8], and designed minimum distance d ≥ |I|+1 = 10 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(764, large, F7, 9) (dual of [large, large−64, 10]-code), using
(65−9, 65, large)-Net in Base 7 — Upper bound on s
There is no (56, 65, large)-net in base 7, because
- 7 times m-reduction [i] would yield (56, 58, large)-net in base 7, but