Best Known (31, 31+3, s)-Nets in Base 7
(31, 31+3, large)-Net over F7 — Constructive and digital
Digital (31, 34, large)-net over F7, using
- t-expansion [i] based on digital (30, 34, large)-net over F7, using
- net defined by OOA [i] based on linear OOA(734, large, F7, 4, 4), using
- appending kth column [i] based on linear OOA(734, large, F7, 3, 4), using
- (u, u+v)-construction [i] based on
- linear OOA(72, 8, F7, 3, 2) (dual of [(8, 3), 22, 3]-NRT-code), using
- extended Reed–Solomon NRT-code RSe(3;22,7) [i]
- linear OOA(732, 8388602, F7, 3, 4) (dual of [(8388602, 3), 25165774, 5]-NRT-code), using
- trace code [i] based on linear OOA(4916, 4194301, F49, 3, 4) (dual of [(4194301, 3), 12582887, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(4916, 8388602, F49, 4) (dual of [8388602, 8388586, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(4916, large, F49, 4) (dual of [large, large−16, 5]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 25679568 | 495−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 5 [i]
- discarding factors / shortening the dual code based on linear OA(4916, large, F49, 4) (dual of [large, large−16, 5]-code), using
- OA 2-folding and stacking [i] based on linear OA(4916, 8388602, F49, 4) (dual of [8388602, 8388586, 5]-code), using
- trace code [i] based on linear OOA(4916, 4194301, F49, 3, 4) (dual of [(4194301, 3), 12582887, 5]-NRT-code), using
- linear OOA(72, 8, F7, 3, 2) (dual of [(8, 3), 22, 3]-NRT-code), using
- (u, u+v)-construction [i] based on
- appending kth column [i] based on linear OOA(734, large, F7, 3, 4), using
- net defined by OOA [i] based on linear OOA(734, large, F7, 4, 4), using
(31, 31+3, large)-Net in Base 7 — Upper bound on s
There is no (31, 34, large)-net in base 7, because
- 1 times m-reduction [i] would yield (31, 33, large)-net in base 7, but