Best Known (27, 27+28, s)-Nets in Base 81
(27, 27+28, 468)-Net over F81 — Constructive and digital
Digital (27, 55, 468)-net over F81, using
- net defined by OOA [i] based on linear OOA(8155, 468, F81, 28, 28) (dual of [(468, 28), 13049, 29]-NRT-code), using
- OA 14-folding and stacking [i] based on linear OA(8155, 6552, F81, 28) (dual of [6552, 6497, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(8155, 6561, F81, 28) (dual of [6561, 6506, 29]-code), using
- an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- discarding factors / shortening the dual code based on linear OA(8155, 6561, F81, 28) (dual of [6561, 6506, 29]-code), using
- OA 14-folding and stacking [i] based on linear OA(8155, 6552, F81, 28) (dual of [6552, 6497, 29]-code), using
(27, 27+28, 1661)-Net over F81 — Digital
Digital (27, 55, 1661)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8155, 1661, F81, 3, 28) (dual of [(1661, 3), 4928, 29]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8155, 2187, F81, 3, 28) (dual of [(2187, 3), 6506, 29]-NRT-code), using
- OOA 3-folding [i] based on linear OA(8155, 6561, F81, 28) (dual of [6561, 6506, 29]-code), using
- an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- OOA 3-folding [i] based on linear OA(8155, 6561, F81, 28) (dual of [6561, 6506, 29]-code), using
- discarding factors / shortening the dual code based on linear OOA(8155, 2187, F81, 3, 28) (dual of [(2187, 3), 6506, 29]-NRT-code), using
(27, 27+28, 2376765)-Net in Base 81 — Upper bound on s
There is no (27, 55, 2376766)-net in base 81, because
- the generalized Rao bound for nets shows that 81m ≥ 926 143979 964732 993115 730005 416024 671902 711601 270706 249984 998745 156283 651039 122942 262237 123083 260521 632321 > 8155 [i]