Best Known (6, 10, s)-Nets in Base 27
(6, 10, 9843)-Net over F27 — Constructive and digital
Digital (6, 10, 9843)-net over F27, using
- net defined by OOA [i] based on linear OOA(2710, 9843, F27, 4, 4) (dual of [(9843, 4), 39362, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(2710, 19686, F27, 4) (dual of [19686, 19676, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- linear OA(2710, 19683, F27, 4) (dual of [19683, 19673, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(277, 19683, F27, 3) (dual of [19683, 19676, 4]-code or 19683-cap in PG(6,27)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(270, 3, F27, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(270, s, F27, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- OA 2-folding and stacking [i] based on linear OA(2710, 19686, F27, 4) (dual of [19686, 19676, 5]-code), using
(6, 10, 19686)-Net over F27 — Digital
Digital (6, 10, 19686)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2710, 19686, F27, 4) (dual of [19686, 19676, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- linear OA(2710, 19683, F27, 4) (dual of [19683, 19673, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(277, 19683, F27, 3) (dual of [19683, 19676, 4]-code or 19683-cap in PG(6,27)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(270, 3, F27, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(270, s, F27, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
(6, 10, 780477)-Net in Base 27 — Upper bound on s
There is no (6, 10, 780478)-net in base 27, because
- the generalized Rao bound for nets shows that 27m ≥ 205 891621 454013 > 2710 [i]