Best Known (57−6, 57, s)-Nets in Base 7
(57−6, 57, 7686412)-Net over F7 — Constructive and digital
Digital (51, 57, 7686412)-net over F7, using
- (u, u+v)-construction [i] 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
- digital (38, 44, 3843206)-net over F7, using
- trace code for nets [i] based on digital (16, 22, 1921603)-net over F49, using
- net defined by OOA [i] based on linear OOA(4922, 1921603, F49, 6, 6) (dual of [(1921603, 6), 11529596, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(4922, 5764809, F49, 6) (dual of [5764809, 5764787, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(4922, 5764810, F49, 6) (dual of [5764810, 5764788, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(3) [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(4913, 5764801, F49, 4) (dual of [5764801, 5764788, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 494−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(491, 9, F49, 1) (dual of [9, 8, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(491, s, F49, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(5) ⊂ Ce(3) [i] based on
- discarding factors / shortening the dual code based on linear OA(4922, 5764810, F49, 6) (dual of [5764810, 5764788, 7]-code), using
- OA 3-folding and stacking [i] based on linear OA(4922, 5764809, F49, 6) (dual of [5764809, 5764787, 7]-code), using
- net defined by OOA [i] based on linear OOA(4922, 1921603, F49, 6, 6) (dual of [(1921603, 6), 11529596, 7]-NRT-code), using
- trace code for nets [i] based on digital (16, 22, 1921603)-net over F49, using
- digital (10, 13, 6120101)-net over F7, using
(57−6, 57, large)-Net over F7 — Digital
Digital (51, 57, large)-net over F7, using
- 72 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
(57−6, 57, large)-Net in Base 7 — Upper bound on s
There is no (51, 57, large)-net in base 7, because
- 4 times m-reduction [i] would yield (51, 53, large)-net in base 7, but