Best Known (57, 57+31, s)-Nets in Base 27
(57, 57+31, 1312)-Net over F27 — Constructive and digital
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
(57, 57+31, 9843)-Net over F27 — Digital
Digital (57, 88, 9843)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2788, 9843, F27, 2, 31) (dual of [(9843, 2), 19598, 32]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2788, 19686, F27, 31) (dual of [19686, 19598, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(29) [i] 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]
- 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(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(30) ⊂ Ce(29) [i] based on
- OOA 2-folding [i] based on linear OA(2788, 19686, F27, 31) (dual of [19686, 19598, 32]-code), using
(57, 57+31, large)-Net in Base 27 — Upper bound on s
There is no (57, 88, large)-net in base 27, because
- 29 times m-reduction [i] would yield (57, 59, large)-net in base 27, but