Best Known (86−30, 86, s)-Nets in Base 27
(86−30, 86, 1312)-Net over F27 — Constructive and digital
Digital (56, 86, 1312)-net over F27, using
- 271 times duplication [i] based on digital (55, 85, 1312)-net over F27, using
- net defined by OOA [i] based on linear OOA(2785, 1312, F27, 30, 30) (dual of [(1312, 30), 39275, 31]-NRT-code), using
- OA 15-folding and stacking [i] based on linear OA(2785, 19680, F27, 30) (dual of [19680, 19595, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(2785, 19683, F27, 30) (dual of [19683, 19598, 31]-code), using
- an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- discarding factors / shortening the dual code based on linear OA(2785, 19683, F27, 30) (dual of [19683, 19598, 31]-code), using
- OA 15-folding and stacking [i] based on linear OA(2785, 19680, F27, 30) (dual of [19680, 19595, 31]-code), using
- net defined by OOA [i] based on linear OOA(2785, 1312, F27, 30, 30) (dual of [(1312, 30), 39275, 31]-NRT-code), using
(86−30, 86, 9845)-Net over F27 — Digital
Digital (56, 86, 9845)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2786, 9845, F27, 2, 30) (dual of [(9845, 2), 19604, 31]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2786, 19690, F27, 30) (dual of [19690, 19604, 31]-code), using
- construction X applied to Ce(29) ⊂ Ce(27) [i] based on
- linear OA(2785, 19683, F27, 30) (dual of [19683, 19598, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(2779, 19683, F27, 28) (dual of [19683, 19604, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(271, 7, F27, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(271, s, F27, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(29) ⊂ Ce(27) [i] based on
- OOA 2-folding [i] based on linear OA(2786, 19690, F27, 30) (dual of [19690, 19604, 31]-code), using
(86−30, 86, large)-Net in Base 27 — Upper bound on s
There is no (56, 86, large)-net in base 27, because
- 28 times m-reduction [i] would yield (56, 58, large)-net in base 27, but