Best Known (67−11, 67, s)-Nets in Base 7
(67−11, 67, 164713)-Net over F7 — Constructive and digital
Digital (56, 67, 164713)-net over F7, using
- net defined by OOA [i] based on linear OOA(767, 164713, F7, 11, 11) (dual of [(164713, 11), 1811776, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(767, 823566, F7, 11) (dual of [823566, 823499, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(767, 823567, F7, 11) (dual of [823567, 823500, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(7) [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(743, 823543, F7, 8) (dual of [823543, 823500, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(73, 24, F7, 2) (dual of [24, 21, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(73, 48, F7, 2) (dual of [48, 45, 3]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 48 = 72−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- discarding factors / shortening the dual code based on linear OA(73, 48, F7, 2) (dual of [48, 45, 3]-code), using
- construction X applied to Ce(10) ⊂ Ce(7) [i] based on
- discarding factors / shortening the dual code based on linear OA(767, 823567, F7, 11) (dual of [823567, 823500, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(767, 823566, F7, 11) (dual of [823566, 823499, 12]-code), using
(67−11, 67, 823567)-Net over F7 — Digital
Digital (56, 67, 823567)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(767, 823567, F7, 11) (dual of [823567, 823500, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(7) [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(743, 823543, F7, 8) (dual of [823543, 823500, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(73, 24, F7, 2) (dual of [24, 21, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(73, 48, F7, 2) (dual of [48, 45, 3]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 48 = 72−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- discarding factors / shortening the dual code based on linear OA(73, 48, F7, 2) (dual of [48, 45, 3]-code), using
- construction X applied to Ce(10) ⊂ Ce(7) [i] based on
(67−11, 67, large)-Net in Base 7 — Upper bound on s
There is no (56, 67, large)-net in base 7, because
- 9 times m-reduction [i] would yield (56, 58, large)-net in base 7, but