Best Known (34−8, 34, s)-Nets in Base 7
(34−8, 34, 4205)-Net over F7 — Constructive and digital
Digital (26, 34, 4205)-net over F7, using
- net defined by OOA [i] based on linear OOA(734, 4205, F7, 8, 8) (dual of [(4205, 8), 33606, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(734, 16820, F7, 8) (dual of [16820, 16786, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(4) [i] based on
- linear OA(731, 16807, F7, 8) (dual of [16807, 16776, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(721, 16807, F7, 5) (dual of [16807, 16786, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(73, 13, F7, 2) (dual of [13, 10, 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(7) ⊂ Ce(4) [i] based on
- OA 4-folding and stacking [i] based on linear OA(734, 16820, F7, 8) (dual of [16820, 16786, 9]-code), using
(34−8, 34, 16820)-Net over F7 — Digital
Digital (26, 34, 16820)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(734, 16820, F7, 8) (dual of [16820, 16786, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(4) [i] based on
- linear OA(731, 16807, F7, 8) (dual of [16807, 16776, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(721, 16807, F7, 5) (dual of [16807, 16786, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(73, 13, F7, 2) (dual of [13, 10, 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(7) ⊂ Ce(4) [i] based on
(34−8, 34, 5626453)-Net in Base 7 — Upper bound on s
There is no (26, 34, 5626454)-net in base 7, because
- the generalized Rao bound for nets shows that 7m ≥ 54116 975800 941953 388474 150697 > 734 [i]