Best Known (56, 56+29, s)-Nets in Base 27
(56, 56+29, 1406)-Net over F27 — Constructive and digital
Digital (56, 85, 1406)-net over F27, using
- 273 times duplication [i] based on digital (53, 82, 1406)-net over F27, using
- net defined by OOA [i] based on linear OOA(2782, 1406, F27, 29, 29) (dual of [(1406, 29), 40692, 30]-NRT-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(2782, 19685, F27, 29) (dual of [19685, 19603, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(2782, 19686, F27, 29) (dual of [19686, 19604, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(27) [i] based on
- 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(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(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(28) ⊂ Ce(27) [i] based on
- discarding factors / shortening the dual code based on linear OA(2782, 19686, F27, 29) (dual of [19686, 19604, 30]-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(2782, 19685, F27, 29) (dual of [19685, 19603, 30]-code), using
- net defined by OOA [i] based on linear OOA(2782, 1406, F27, 29, 29) (dual of [(1406, 29), 40692, 30]-NRT-code), using
(56, 56+29, 11914)-Net over F27 — Digital
Digital (56, 85, 11914)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2785, 11914, F27, 29) (dual of [11914, 11829, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(2785, 19684, F27, 29) (dual of [19684, 19599, 30]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 19684 | 276−1, defining interval I = [0,14], and minimum distance d ≥ |{−14,−13,…,14}|+1 = 30 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(2785, 19684, F27, 29) (dual of [19684, 19599, 30]-code), using
(56, 56+29, large)-Net in Base 27 — Upper bound on s
There is no (56, 85, large)-net in base 27, because
- 27 times m-reduction [i] would yield (56, 58, large)-net in base 27, but