Best Known (58−6, 58, s)-Nets in Base 7
(58−6, 58, 7686461)-Net over F7 — Constructive and digital
Digital (52, 58, 7686461)-net over F7, using
- generalized (u, u+v)-construction [i] based on
- digital (1, 3, 57)-net over F7, using
- digital (10, 13, 3843202)-net over F7, using
- s-reduction based on digital (10, 13, 6120101)-net over F7, using
- net defined by OOA [i] based on linear OOA(713, 6120101, F7, 3, 3) (dual of [(6120101, 3), 18360290, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(713, 6120101, F7, 2, 3) (dual of [(6120101, 2), 12240189, 4]-NRT-code), using
- OAs with strength 3, b ≠ 2, and m > 3 are always embeddable [i] based on linear OA(713, 6120101, F7, 3) (dual of [6120101, 6120088, 4]-code or 6120101-cap in PG(12,7)), using
- appending kth column [i] based on linear OOA(713, 6120101, F7, 2, 3) (dual of [(6120101, 2), 12240189, 4]-NRT-code), using
- net defined by OOA [i] based on linear OOA(713, 6120101, F7, 3, 3) (dual of [(6120101, 3), 18360290, 4]-NRT-code), using
- s-reduction based on digital (10, 13, 6120101)-net over F7, 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
(58−6, 58, large)-Net over F7 — Digital
Digital (52, 58, large)-net over F7, using
- 73 times duplication [i] based on digital (49, 55, large)-net over F7, using
- t-expansion [i] based on digital (47, 55, large)-net over F7, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(755, large, F7, 8) (dual of [large, large−55, 9]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 20176803 | 79−1, defining interval I = [0,7], and designed minimum distance d ≥ |I|+1 = 9 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(755, large, F7, 8) (dual of [large, large−55, 9]-code), using
- t-expansion [i] based on digital (47, 55, large)-net over F7, using
(58−6, 58, large)-Net in Base 7 — Upper bound on s
There is no (52, 58, large)-net in base 7, because
- 4 times m-reduction [i] would yield (52, 54, large)-net in base 7, but