Best Known (102−9, 102, s)-Nets in Base 7
(102−9, 102, 6291450)-Net over F7 — Constructive and digital
Digital (93, 102, 6291450)-net over F7, using
- generalized (u, u+v)-construction [i] based on
- digital (10, 13, 2097150)-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 (21, 25, 2097150)-net over F7, using
- s-reduction based on digital (21, 25, 2882404)-net over F7, using
- net defined by OOA [i] based on linear OOA(725, 2882404, F7, 4, 4) (dual of [(2882404, 4), 11529591, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(725, 2882404, F7, 3, 4) (dual of [(2882404, 3), 8647187, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(725, 5764808, F7, 4) (dual of [5764808, 5764783, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(725, 5764809, F7, 4) (dual of [5764809, 5764784, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- linear OA(725, 5764801, F7, 4) (dual of [5764801, 5764776, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 78−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(717, 5764801, F7, 3) (dual of [5764801, 5764784, 4]-code or 5764801-cap in PG(16,7)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 78−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(70, 8, F7, 0) (dual of [8, 8, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(70, s, F7, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- discarding factors / shortening the dual code based on linear OA(725, 5764809, F7, 4) (dual of [5764809, 5764784, 5]-code), using
- OA 2-folding and stacking [i] based on linear OA(725, 5764808, F7, 4) (dual of [5764808, 5764783, 5]-code), using
- appending kth column [i] based on linear OOA(725, 2882404, F7, 3, 4) (dual of [(2882404, 3), 8647187, 5]-NRT-code), using
- net defined by OOA [i] based on linear OOA(725, 2882404, F7, 4, 4) (dual of [(2882404, 4), 11529591, 5]-NRT-code), using
- s-reduction based on digital (21, 25, 2882404)-net over F7, using
- digital (55, 64, 2097150)-net over F7, using
- net defined by OOA [i] based on linear OOA(764, 2097150, F7, 9, 9) (dual of [(2097150, 9), 18874286, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(764, 8388601, F7, 9) (dual of [8388601, 8388537, 10]-code), using
- discarding factors / shortening the dual code 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]
- discarding factors / shortening the dual code based on linear OA(764, large, F7, 9) (dual of [large, large−64, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(764, 8388601, F7, 9) (dual of [8388601, 8388537, 10]-code), using
- net defined by OOA [i] based on linear OOA(764, 2097150, F7, 9, 9) (dual of [(2097150, 9), 18874286, 10]-NRT-code), using
- digital (10, 13, 2097150)-net over F7, using
(102−9, 102, large)-Net over F7 — Digital
Digital (93, 102, large)-net over F7, using
- t-expansion [i] based on digital (92, 102, large)-net over F7, using
- 4 times m-reduction [i] based on digital (92, 106, large)-net over F7, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(7106, large, F7, 14) (dual of [large, large−106, 15]-code), using
- trace code [i] based on linear OA(4953, 5764801, F49, 14) (dual of [5764801, 5764748, 15]-code), using
- an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 494−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- trace code [i] based on linear OA(4953, 5764801, F49, 14) (dual of [5764801, 5764748, 15]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(7106, large, F7, 14) (dual of [large, large−106, 15]-code), using
- 4 times m-reduction [i] based on digital (92, 106, large)-net over F7, using
(102−9, 102, large)-Net in Base 7 — Upper bound on s
There is no (93, 102, large)-net in base 7, because
- 7 times m-reduction [i] would yield (93, 95, large)-net in base 7, but