Best Known (90−31, 90, s)-Nets in Base 27
(90−31, 90, 1312)-Net over F27 — Constructive and digital
Digital (59, 90, 1312)-net over F27, using
- 272 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
(90−31, 90, 11076)-Net over F27 — Digital
Digital (59, 90, 11076)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2790, 11076, F27, 31) (dual of [11076, 10986, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(2790, 19694, F27, 31) (dual of [19694, 19604, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(27) [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(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(272, 11, F27, 2) (dual of [11, 9, 3]-code or 11-arc in PG(1,27)), using
- discarding factors / shortening the dual code based on linear OA(272, 27, F27, 2) (dual of [27, 25, 3]-code or 27-arc in PG(1,27)), using
- Reed–Solomon code RS(25,27) [i]
- discarding factors / shortening the dual code based on linear OA(272, 27, F27, 2) (dual of [27, 25, 3]-code or 27-arc in PG(1,27)), using
- construction X applied to Ce(30) ⊂ Ce(27) [i] based on
- discarding factors / shortening the dual code based on linear OA(2790, 19694, F27, 31) (dual of [19694, 19604, 32]-code), using
(90−31, 90, large)-Net in Base 27 — Upper bound on s
There is no (59, 90, large)-net in base 27, because
- 29 times m-reduction [i] would yield (59, 61, large)-net in base 27, but