Best Known (71−11, 71, s)-Nets in Base 7
(71−11, 71, 164730)-Net over F7 — Constructive and digital
Digital (60, 71, 164730)-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 (53, 64, 164709)-net over F7, using
- net defined by OOA [i] based on linear OOA(764, 164709, F7, 11, 11) (dual of [(164709, 11), 1811735, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(764, 823546, F7, 11) (dual of [823546, 823482, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(764, 823550, F7, 11) (dual of [823550, 823486, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- linear OA(764, 823543, F7, 11) (dual of [823543, 823479, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(757, 823543, F7, 10) (dual of [823543, 823486, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(70, 7, F7, 0) (dual of [7, 7, 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(764, 823550, F7, 11) (dual of [823550, 823486, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(764, 823546, F7, 11) (dual of [823546, 823482, 12]-code), using
- net defined by OOA [i] based on linear OOA(764, 164709, F7, 11, 11) (dual of [(164709, 11), 1811735, 12]-NRT-code), using
- digital (2, 7, 21)-net over F7, using
(71−11, 71, 823593)-Net over F7 — Digital
Digital (60, 71, 823593)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(771, 823593, F7, 11) (dual of [823593, 823522, 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(764, 823550, F7, 11) (dual of [823550, 823486, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- linear OA(764, 823543, F7, 11) (dual of [823543, 823479, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(757, 823543, F7, 10) (dual of [823543, 823486, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(70, 7, F7, 0) (dual of [7, 7, 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
(71−11, 71, large)-Net in Base 7 — Upper bound on s
There is no (60, 71, large)-net in base 7, because
- 9 times m-reduction [i] would yield (60, 62, large)-net in base 7, but