Best Known (78, 78+21, s)-Nets in Base 7
(78, 78+21, 1684)-Net over F7 — Constructive and digital
Digital (78, 99, 1684)-net over F7, using
- 71 times duplication [i] based on digital (77, 98, 1684)-net over F7, using
- net defined by OOA [i] based on linear OOA(798, 1684, F7, 21, 21) (dual of [(1684, 21), 35266, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(798, 16841, F7, 21) (dual of [16841, 16743, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(798, 16845, F7, 21) (dual of [16845, 16747, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,7]) [i] based on
- linear OA(791, 16808, F7, 21) (dual of [16808, 16717, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 16808 | 710−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(761, 16808, F7, 15) (dual of [16808, 16747, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 16808 | 710−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(77, 37, F7, 5) (dual of [37, 30, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(77, 43, F7, 5) (dual of [43, 36, 6]-code), using
- construction X applied to C([0,10]) ⊂ C([0,7]) [i] based on
- discarding factors / shortening the dual code based on linear OA(798, 16845, F7, 21) (dual of [16845, 16747, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(798, 16841, F7, 21) (dual of [16841, 16743, 22]-code), using
- net defined by OOA [i] based on linear OOA(798, 1684, F7, 21, 21) (dual of [(1684, 21), 35266, 22]-NRT-code), using
(78, 78+21, 21115)-Net over F7 — Digital
Digital (78, 99, 21115)-net over F7, using
(78, 78+21, large)-Net in Base 7 — Upper bound on s
There is no (78, 99, large)-net in base 7, because
- 19 times m-reduction [i] would yield (78, 80, large)-net in base 7, but