Best Known (69, 69+9, s)-Nets in Base 7
(69, 69+9, 2884754)-Net over F7 — Constructive and digital
Digital (69, 78, 2884754)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (8, 12, 2352)-net over F7, using
- net defined by OOA [i] based on linear OOA(712, 2352, F7, 4, 4) (dual of [(2352, 4), 9396, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(712, 2352, F7, 3, 4) (dual of [(2352, 3), 7044, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(712, 4704, F7, 4) (dual of [4704, 4692, 5]-code), using
- trace code [i] based on linear OA(496, 2352, F49, 4) (dual of [2352, 2346, 5]-code), using
- 1 times truncation [i] based on linear OA(497, 2353, F49, 5) (dual of [2353, 2346, 6]-code), using
- trace code [i] based on linear OA(496, 2352, F49, 4) (dual of [2352, 2346, 5]-code), using
- OA 2-folding and stacking [i] based on linear OA(712, 4704, F7, 4) (dual of [4704, 4692, 5]-code), using
- appending kth column [i] based on linear OOA(712, 2352, F7, 3, 4) (dual of [(2352, 3), 7044, 5]-NRT-code), using
- net defined by OOA [i] based on linear OOA(712, 2352, F7, 4, 4) (dual of [(2352, 4), 9396, 5]-NRT-code), using
- digital (57, 66, 2882402)-net over F7, using
- trace code for nets [i] based on digital (24, 33, 1441201)-net over F49, using
- net defined by OOA [i] based on linear OOA(4933, 1441201, F49, 9, 9) (dual of [(1441201, 9), 12970776, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(4933, 5764805, F49, 9) (dual of [5764805, 5764772, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(7) [i] based on
- linear OA(4933, 5764801, F49, 9) (dual of [5764801, 5764768, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 494−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(4929, 5764801, F49, 8) (dual of [5764801, 5764772, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 494−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(490, 4, F49, 0) (dual of [4, 4, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(490, s, F49, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(8) ⊂ Ce(7) [i] based on
- OOA 4-folding and stacking with additional row [i] based on linear OA(4933, 5764805, F49, 9) (dual of [5764805, 5764772, 10]-code), using
- net defined by OOA [i] based on linear OOA(4933, 1441201, F49, 9, 9) (dual of [(1441201, 9), 12970776, 10]-NRT-code), using
- trace code for nets [i] based on digital (24, 33, 1441201)-net over F49, using
- digital (8, 12, 2352)-net over F7, using
(69, 69+9, large)-Net over F7 — Digital
Digital (69, 78, large)-net over F7, using
- 75 times duplication [i] based on digital (64, 73, large)-net over F7, using
- t-expansion [i] based on digital (63, 73, large)-net over F7, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(773, large, F7, 10) (dual of [large, large−73, 11]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 20176803 | 79−1, defining interval I = [0,9], and designed minimum distance d ≥ |I|+1 = 11 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(773, large, F7, 10) (dual of [large, large−73, 11]-code), using
- t-expansion [i] based on digital (63, 73, large)-net over F7, using
(69, 69+9, large)-Net in Base 7 — Upper bound on s
There is no (69, 78, large)-net in base 7, because
- 7 times m-reduction [i] would yield (69, 71, large)-net in base 7, but