Best Known (62, 62+8, s)-Nets in Base 7
(62, 62+8, 2884754)-Net over F7 — Constructive and digital
Digital (62, 70, 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 (50, 58, 2882402)-net over F7, using
- trace code for nets [i] based on digital (21, 29, 1441201)-net over F49, using
- net defined by OOA [i] based on linear OOA(4929, 1441201, F49, 8, 8) (dual of [(1441201, 8), 11529579, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(4929, 5764804, F49, 8) (dual of [5764804, 5764775, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(4929, 5764805, F49, 8) (dual of [5764805, 5764776, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(6) [i] based on
- 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(4925, 5764801, F49, 7) (dual of [5764801, 5764776, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 494−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [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(7) ⊂ Ce(6) [i] based on
- discarding factors / shortening the dual code based on linear OA(4929, 5764805, F49, 8) (dual of [5764805, 5764776, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(4929, 5764804, F49, 8) (dual of [5764804, 5764775, 9]-code), using
- net defined by OOA [i] based on linear OOA(4929, 1441201, F49, 8, 8) (dual of [(1441201, 8), 11529579, 9]-NRT-code), using
- trace code for nets [i] based on digital (21, 29, 1441201)-net over F49, using
- digital (8, 12, 2352)-net over F7, using
(62, 62+8, large)-Net over F7 — Digital
Digital (62, 70, large)-net over F7, using
- 76 times duplication [i] based on digital (56, 64, large)-net over F7, using
- t-expansion [i] based on digital (55, 64, large)-net over F7, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(764, large, F7, 9) (dual of [large, large−64, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 20176803 | 79−1, defining interval I = [0,8], and designed minimum distance d ≥ |I|+1 = 10 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(764, large, F7, 9) (dual of [large, large−64, 10]-code), using
- t-expansion [i] based on digital (55, 64, large)-net over F7, using
(62, 62+8, large)-Net in Base 7 — Upper bound on s
There is no (62, 70, large)-net in base 7, because
- 6 times m-reduction [i] would yield (62, 64, large)-net in base 7, but