Best Known (51−6, 51, s)-Nets in Base 7
(51−6, 51, 3864757)-Net over F7 — Constructive and digital
Digital (45, 51, 3864757)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (6, 9, 21555)-net over F7, using
- net defined by OOA [i] based on linear OOA(79, 21555, F7, 3, 3) (dual of [(21555, 3), 64656, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(79, 21555, F7, 2, 3) (dual of [(21555, 2), 43101, 4]-NRT-code), using
- net defined by OOA [i] based on linear OOA(79, 21555, F7, 3, 3) (dual of [(21555, 3), 64656, 4]-NRT-code), using
- digital (36, 42, 3843202)-net over F7, using
- trace code for nets [i] based on digital (15, 21, 1921601)-net over F49, using
- net defined by OOA [i] based on linear OOA(4921, 1921601, F49, 6, 6) (dual of [(1921601, 6), 11529585, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(4921, 5764803, F49, 6) (dual of [5764803, 5764782, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(4921, 5764805, F49, 6) (dual of [5764805, 5764784, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- linear OA(4921, 5764801, F49, 6) (dual of [5764801, 5764780, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 494−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(4917, 5764801, F49, 5) (dual of [5764801, 5764784, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 494−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [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(5) ⊂ Ce(4) [i] based on
- discarding factors / shortening the dual code based on linear OA(4921, 5764805, F49, 6) (dual of [5764805, 5764784, 7]-code), using
- OA 3-folding and stacking [i] based on linear OA(4921, 5764803, F49, 6) (dual of [5764803, 5764782, 7]-code), using
- net defined by OOA [i] based on linear OOA(4921, 1921601, F49, 6, 6) (dual of [(1921601, 6), 11529585, 7]-NRT-code), using
- trace code for nets [i] based on digital (15, 21, 1921601)-net over F49, using
- digital (6, 9, 21555)-net over F7, using
(51−6, 51, large)-Net over F7 — Digital
Digital (45, 51, large)-net over F7, using
- 71 times duplication [i] based on digital (44, 50, large)-net over F7, using
- t-expansion [i] based on digital (43, 50, large)-net over F7, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(750, large, F7, 7) (dual of [large, large−50, 8]-code), using
- trace code [i] based on linear OA(4925, 5764802, F49, 7) (dual of [5764802, 5764777, 8]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 5764802 | 498−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- trace code [i] based on linear OA(4925, 5764802, F49, 7) (dual of [5764802, 5764777, 8]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(750, large, F7, 7) (dual of [large, large−50, 8]-code), using
- t-expansion [i] based on digital (43, 50, large)-net over F7, using
(51−6, 51, large)-Net in Base 7 — Upper bound on s
There is no (45, 51, large)-net in base 7, because
- 4 times m-reduction [i] would yield (45, 47, large)-net in base 7, but