Best Known (26−8, 26, s)-Nets in Base 7
(26−8, 26, 601)-Net over F7 — Constructive and digital
Digital (18, 26, 601)-net over F7, using
- net defined by OOA [i] based on linear OOA(726, 601, F7, 8, 8) (dual of [(601, 8), 4782, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(726, 2404, F7, 8) (dual of [2404, 2378, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(726, 2406, F7, 8) (dual of [2406, 2380, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(5) [i] based on
- linear OA(725, 2401, F7, 8) (dual of [2401, 2376, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(721, 2401, F7, 6) (dual of [2401, 2380, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(71, 5, F7, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(71, s, F7, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(7) ⊂ Ce(5) [i] based on
- discarding factors / shortening the dual code based on linear OA(726, 2406, F7, 8) (dual of [2406, 2380, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(726, 2404, F7, 8) (dual of [2404, 2378, 9]-code), using
(26−8, 26, 1654)-Net over F7 — Digital
Digital (18, 26, 1654)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(726, 1654, F7, 8) (dual of [1654, 1628, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(726, 2406, F7, 8) (dual of [2406, 2380, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(5) [i] based on
- linear OA(725, 2401, F7, 8) (dual of [2401, 2376, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(721, 2401, F7, 6) (dual of [2401, 2380, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(71, 5, F7, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(71, s, F7, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(7) ⊂ Ce(5) [i] based on
- discarding factors / shortening the dual code based on linear OA(726, 2406, F7, 8) (dual of [2406, 2380, 9]-code), using
(26−8, 26, 114823)-Net in Base 7 — Upper bound on s
There is no (18, 26, 114824)-net in base 7, because
- the generalized Rao bound for nets shows that 7m ≥ 9387 657281 191897 603777 > 726 [i]