Best Known (51, 77, s)-Nets in Base 27
(51, 77, 1514)-Net over F27 — Constructive and digital
Digital (51, 77, 1514)-net over F27, using
- 271 times duplication [i] based on digital (50, 76, 1514)-net over F27, using
- net defined by OOA [i] based on linear OOA(2776, 1514, F27, 26, 26) (dual of [(1514, 26), 39288, 27]-NRT-code), using
- OA 13-folding and stacking [i] based on linear OA(2776, 19682, F27, 26) (dual of [19682, 19606, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(2776, 19683, F27, 26) (dual of [19683, 19607, 27]-code), using
- an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- discarding factors / shortening the dual code based on linear OA(2776, 19683, F27, 26) (dual of [19683, 19607, 27]-code), using
- OA 13-folding and stacking [i] based on linear OA(2776, 19682, F27, 26) (dual of [19682, 19606, 27]-code), using
- net defined by OOA [i] based on linear OOA(2776, 1514, F27, 26, 26) (dual of [(1514, 26), 39288, 27]-NRT-code), using
(51, 77, 12842)-Net over F27 — Digital
Digital (51, 77, 12842)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2777, 12842, F27, 26) (dual of [12842, 12765, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(2777, 19690, F27, 26) (dual of [19690, 19613, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(23) [i] based on
- linear OA(2776, 19683, F27, 26) (dual of [19683, 19607, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(2770, 19683, F27, 24) (dual of [19683, 19613, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [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(25) ⊂ Ce(23) [i] based on
- discarding factors / shortening the dual code based on linear OA(2777, 19690, F27, 26) (dual of [19690, 19613, 27]-code), using
(51, 77, large)-Net in Base 27 — Upper bound on s
There is no (51, 77, large)-net in base 27, because
- 24 times m-reduction [i] would yield (51, 53, large)-net in base 27, but