Best Known (47, 56, s)-Nets in Base 7
(47, 56, 205908)-Net over F7 — Constructive and digital
Digital (47, 56, 205908)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (2, 6, 21)-net over F7, using
- net defined by OOA [i] based on linear OOA(76, 21, F7, 4, 4) (dual of [(21, 4), 78, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(76, 21, F7, 3, 4) (dual of [(21, 3), 57, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(76, 42, F7, 4) (dual of [42, 36, 5]-code), using
- 1 times truncation [i] based on linear OA(77, 43, F7, 5) (dual of [43, 36, 6]-code), using
- OA 2-folding and stacking [i] based on linear OA(76, 42, F7, 4) (dual of [42, 36, 5]-code), using
- appending kth column [i] based on linear OOA(76, 21, F7, 3, 4) (dual of [(21, 3), 57, 5]-NRT-code), using
- net defined by OOA [i] based on linear OOA(76, 21, F7, 4, 4) (dual of [(21, 4), 78, 5]-NRT-code), using
- digital (41, 50, 205887)-net over F7, using
- net defined by OOA [i] based on linear OOA(750, 205887, F7, 9, 9) (dual of [(205887, 9), 1852933, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(750, 823549, F7, 9) (dual of [823549, 823499, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(750, 823550, F7, 9) (dual of [823550, 823500, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(7) [i] based on
- linear OA(750, 823543, F7, 9) (dual of [823543, 823493, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(743, 823543, F7, 8) (dual of [823543, 823500, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(70, 7, F7, 0) (dual of [7, 7, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(70, s, F7, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(8) ⊂ Ce(7) [i] based on
- discarding factors / shortening the dual code based on linear OA(750, 823550, F7, 9) (dual of [823550, 823500, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(750, 823549, F7, 9) (dual of [823549, 823499, 10]-code), using
- net defined by OOA [i] based on linear OOA(750, 205887, F7, 9, 9) (dual of [(205887, 9), 1852933, 10]-NRT-code), using
- digital (2, 6, 21)-net over F7, using
(47, 56, 823594)-Net over F7 — Digital
Digital (47, 56, 823594)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(756, 823594, F7, 9) (dual of [823594, 823538, 10]-code), using
- (u, u+v)-construction [i] based on
- linear OA(76, 44, F7, 4) (dual of [44, 38, 5]-code), using
- a “GraX†code from Grassl’s database [i]
- linear OA(750, 823550, F7, 9) (dual of [823550, 823500, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(7) [i] based on
- linear OA(750, 823543, F7, 9) (dual of [823543, 823493, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(743, 823543, F7, 8) (dual of [823543, 823500, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(70, 7, F7, 0) (dual of [7, 7, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(70, s, F7, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(8) ⊂ Ce(7) [i] based on
- linear OA(76, 44, F7, 4) (dual of [44, 38, 5]-code), using
- (u, u+v)-construction [i] based on
(47, 56, large)-Net in Base 7 — Upper bound on s
There is no (47, 56, large)-net in base 7, because
- 7 times m-reduction [i] would yield (47, 49, large)-net in base 7, but