Best Known (89−31, 89, s)-Nets in Base 27
(89−31, 89, 1312)-Net over F27 — Constructive and digital
Digital (58, 89, 1312)-net over F27, using
- 271 times duplication [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
(89−31, 89, 9885)-Net over F27 — Digital
Digital (58, 89, 9885)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2789, 9885, F27, 31) (dual of [9885, 9796, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(2789, 19690, F27, 31) (dual of [19690, 19601, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(28) [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(2782, 19683, F27, 29) (dual of [19683, 19601, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [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(30) ⊂ Ce(28) [i] based on
- discarding factors / shortening the dual code based on linear OA(2789, 19690, F27, 31) (dual of [19690, 19601, 32]-code), using
(89−31, 89, large)-Net in Base 27 — Upper bound on s
There is no (58, 89, large)-net in base 27, because
- 29 times m-reduction [i] would yield (58, 60, large)-net in base 27, but