Best Known (57, 57+30, s)-Nets in Base 27
(57, 57+30, 1312)-Net over F27 — Constructive and digital
Digital (57, 87, 1312)-net over F27, using
- 1 times m-reduction [i] based on digital (57, 88, 1312)-net over F27, using
- net defined by OOA [i] based on linear OOA(2788, 1312, F27, 31, 31) (dual of [(1312, 31), 40584, 32]-NRT-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(2788, 19681, F27, 31) (dual of [19681, 19593, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(2788, 19683, F27, 31) (dual of [19683, 19595, 32]-code), using
- an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- discarding factors / shortening the dual code based on linear OA(2788, 19683, F27, 31) (dual of [19683, 19595, 32]-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(2788, 19681, F27, 31) (dual of [19681, 19593, 32]-code), using
- net defined by OOA [i] based on linear OOA(2788, 1312, F27, 31, 31) (dual of [(1312, 31), 40584, 32]-NRT-code), using
(57, 57+30, 10809)-Net over F27 — Digital
Digital (57, 87, 10809)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2787, 10809, F27, 30) (dual of [10809, 10722, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(2787, 19691, F27, 30) (dual of [19691, 19604, 31]-code), using
- 1 times code embedding in larger space [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
- 1 times code embedding in larger space [i] based on linear OA(2786, 19690, F27, 30) (dual of [19690, 19604, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(2787, 19691, F27, 30) (dual of [19691, 19604, 31]-code), using
(57, 57+30, large)-Net in Base 27 — Upper bound on s
There is no (57, 87, large)-net in base 27, because
- 28 times m-reduction [i] would yield (57, 59, large)-net in base 27, but