Best Known (69, 80, s)-Nets in Base 7
(69, 80, 1152982)-Net over F7 — Constructive and digital
Digital (69, 80, 1152982)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (2, 7, 21)-net over F7, using
- net defined by OOA [i] based on linear OOA(77, 21, F7, 5, 5) (dual of [(21, 5), 98, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(77, 43, F7, 5) (dual of [43, 36, 6]-code), using
- net defined by OOA [i] based on linear OOA(77, 21, F7, 5, 5) (dual of [(21, 5), 98, 6]-NRT-code), using
- digital (62, 73, 1152961)-net over F7, using
- net defined by OOA [i] based on linear OOA(773, 1152961, F7, 11, 11) (dual of [(1152961, 11), 12682498, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(773, 5764806, F7, 11) (dual of [5764806, 5764733, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(773, 5764809, F7, 11) (dual of [5764809, 5764736, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- 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(765, 5764801, F7, 10) (dual of [5764801, 5764736, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 78−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [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(10) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(773, 5764809, F7, 11) (dual of [5764809, 5764736, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(773, 5764806, F7, 11) (dual of [5764806, 5764733, 12]-code), using
- net defined by OOA [i] based on linear OOA(773, 1152961, F7, 11, 11) (dual of [(1152961, 11), 12682498, 12]-NRT-code), using
- digital (2, 7, 21)-net over F7, using
(69, 80, 5764852)-Net over F7 — Digital
Digital (69, 80, 5764852)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(780, 5764852, F7, 11) (dual of [5764852, 5764772, 12]-code), using
- (u, u+v)-construction [i] based on
- linear OA(77, 43, F7, 5) (dual of [43, 36, 6]-code), using
- linear OA(773, 5764809, F7, 11) (dual of [5764809, 5764736, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- 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(765, 5764801, F7, 10) (dual of [5764801, 5764736, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 78−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [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(10) ⊂ Ce(9) [i] based on
- (u, u+v)-construction [i] based on
(69, 80, large)-Net in Base 7 — Upper bound on s
There is no (69, 80, large)-net in base 7, because
- 9 times m-reduction [i] would yield (69, 71, large)-net in base 7, but