Best Known (84−9, 84, s)-Nets in Base 7
(84−9, 84, 4194301)-Net over F7 — Constructive and digital
Digital (75, 84, 4194301)-net over F7, using
- net defined by OOA [i] based on linear OOA(784, 4194301, F7, 10, 9) (dual of [(4194301, 10), 41942926, 10]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(784, large, F7, 2, 9), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(782, 8388602, F7, 2, 9) (dual of [(8388602, 2), 16777122, 10]-NRT-code), using
- trace code [i] based on linear OOA(4941, 4194301, F49, 2, 9) (dual of [(4194301, 2), 8388561, 10]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4941, 8388602, F49, 9) (dual of [8388602, 8388561, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(4941, large, F49, 9) (dual of [large, large−41, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 28247525 | 4910−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(4941, large, F49, 9) (dual of [large, large−41, 10]-code), using
- OOA 2-folding [i] based on linear OA(4941, 8388602, F49, 9) (dual of [8388602, 8388561, 10]-code), using
- trace code [i] based on linear OOA(4941, 4194301, F49, 2, 9) (dual of [(4194301, 2), 8388561, 10]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(782, 8388602, F7, 2, 9) (dual of [(8388602, 2), 16777122, 10]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(784, large, F7, 2, 9), using
(84−9, 84, large)-Net over F7 — Digital
Digital (75, 84, large)-net over F7, using
- 72 times duplication [i] based on digital (73, 82, large)-net over F7, using
- t-expansion [i] based on digital (71, 82, large)-net over F7, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(782, large, F7, 11) (dual of [large, large−82, 12]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 20176803 | 79−1, defining interval I = [0,10], and designed minimum distance d ≥ |I|+1 = 12 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(782, large, F7, 11) (dual of [large, large−82, 12]-code), using
- t-expansion [i] based on digital (71, 82, large)-net over F7, using
(84−9, 84, large)-Net in Base 7 — Upper bound on s
There is no (75, 84, large)-net in base 7, because
- 7 times m-reduction [i] would yield (75, 77, large)-net in base 7, but