Best Known (51, 51+25, s)-Nets in Base 27
(51, 51+25, 1641)-Net over F27 — Constructive and digital
Digital (51, 76, 1641)-net over F27, using
- 271 times duplication [i] based on digital (50, 75, 1641)-net over F27, using
- net defined by OOA [i] based on linear OOA(2775, 1641, F27, 25, 25) (dual of [(1641, 25), 40950, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(2775, 19693, F27, 25) (dual of [19693, 19618, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(2775, 19694, F27, 25) (dual of [19694, 19619, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(21) [i] based on
- linear OA(2773, 19683, F27, 25) (dual of [19683, 19610, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(2764, 19683, F27, 22) (dual of [19683, 19619, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [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(24) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(2775, 19694, F27, 25) (dual of [19694, 19619, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(2775, 19693, F27, 25) (dual of [19693, 19618, 26]-code), using
- net defined by OOA [i] based on linear OOA(2775, 1641, F27, 25, 25) (dual of [(1641, 25), 40950, 26]-NRT-code), using
(51, 51+25, 16853)-Net over F27 — Digital
Digital (51, 76, 16853)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2776, 16853, F27, 25) (dual of [16853, 16777, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(2776, 19699, F27, 25) (dual of [19699, 19623, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,10]) [i] based on
- linear OA(2773, 19684, F27, 25) (dual of [19684, 19611, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 276−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(2761, 19684, F27, 21) (dual of [19684, 19623, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 276−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(273, 15, F27, 3) (dual of [15, 12, 4]-code or 15-arc in PG(2,27) or 15-cap in PG(2,27)), using
- discarding factors / shortening the dual code based on linear OA(273, 27, F27, 3) (dual of [27, 24, 4]-code or 27-arc in PG(2,27) or 27-cap in PG(2,27)), using
- Reed–Solomon code RS(24,27) [i]
- discarding factors / shortening the dual code based on linear OA(273, 27, F27, 3) (dual of [27, 24, 4]-code or 27-arc in PG(2,27) or 27-cap in PG(2,27)), using
- construction X applied to C([0,12]) ⊂ C([0,10]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2776, 19699, F27, 25) (dual of [19699, 19623, 26]-code), using
(51, 51+25, large)-Net in Base 27 — Upper bound on s
There is no (51, 76, large)-net in base 27, because
- 23 times m-reduction [i] would yield (51, 53, large)-net in base 27, but