Best Known (76, 76+12, s)-Nets in Base 7
(76, 76+12, 960814)-Net over F7 — Constructive and digital
Digital (76, 88, 960814)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (1, 7, 13)-net over F7, using
- 6 times m-reduction [i] based on digital (1, 13, 13)-net over F7, using
- digital (69, 81, 960801)-net over F7, using
- net defined by OOA [i] based on linear OOA(781, 960801, F7, 12, 12) (dual of [(960801, 12), 11529531, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(781, 5764806, F7, 12) (dual of [5764806, 5764725, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(781, 5764809, F7, 12) (dual of [5764809, 5764728, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- linear OA(781, 5764801, F7, 12) (dual of [5764801, 5764720, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 78−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(773, 5764801, F7, 11) (dual of [5764801, 5764728, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 78−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [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(11) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(781, 5764809, F7, 12) (dual of [5764809, 5764728, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(781, 5764806, F7, 12) (dual of [5764806, 5764725, 13]-code), using
- net defined by OOA [i] based on linear OOA(781, 960801, F7, 12, 12) (dual of [(960801, 12), 11529531, 13]-NRT-code), using
- digital (1, 7, 13)-net over F7, using
(76, 76+12, 5764844)-Net over F7 — Digital
Digital (76, 88, 5764844)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(788, 5764844, F7, 12) (dual of [5764844, 5764756, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(5) [i] based on
- linear OA(781, 5764801, F7, 12) (dual of [5764801, 5764720, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 78−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(741, 5764801, F7, 6) (dual of [5764801, 5764760, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 78−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(77, 43, F7, 5) (dual of [43, 36, 6]-code), using
- construction X applied to Ce(11) ⊂ Ce(5) [i] based on
(76, 76+12, large)-Net in Base 7 — Upper bound on s
There is no (76, 88, large)-net in base 7, because
- 10 times m-reduction [i] would yield (76, 78, large)-net in base 7, but