Best Known (54, 54+25, s)-Nets in Base 27
(54, 54+25, 1642)-Net over F27 — Constructive and digital
Digital (54, 79, 1642)-net over F27, using
- 271 times duplication [i] based on digital (53, 78, 1642)-net over F27, using
- net defined by OOA [i] based on linear OOA(2778, 1642, F27, 25, 25) (dual of [(1642, 25), 40972, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(2778, 19705, F27, 25) (dual of [19705, 19627, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(2778, 19707, F27, 25) (dual of [19707, 19629, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,9]) [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(2755, 19684, F27, 19) (dual of [19684, 19629, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 276−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(275, 23, F27, 5) (dual of [23, 18, 6]-code or 23-arc in PG(4,27)), using
- discarding factors / shortening the dual code based on linear OA(275, 27, F27, 5) (dual of [27, 22, 6]-code or 27-arc in PG(4,27)), using
- Reed–Solomon code RS(22,27) [i]
- discarding factors / shortening the dual code based on linear OA(275, 27, F27, 5) (dual of [27, 22, 6]-code or 27-arc in PG(4,27)), using
- construction X applied to C([0,12]) ⊂ C([0,9]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2778, 19707, F27, 25) (dual of [19707, 19629, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(2778, 19705, F27, 25) (dual of [19705, 19627, 26]-code), using
- net defined by OOA [i] based on linear OOA(2778, 1642, F27, 25, 25) (dual of [(1642, 25), 40972, 26]-NRT-code), using
(54, 54+25, 19710)-Net over F27 — Digital
Digital (54, 79, 19710)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2779, 19710, F27, 25) (dual of [19710, 19631, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(17) [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(2752, 19683, F27, 18) (dual of [19683, 19631, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(276, 27, F27, 6) (dual of [27, 21, 7]-code or 27-arc in PG(5,27)), using
- Reed–Solomon code RS(21,27) [i]
- construction X applied to Ce(24) ⊂ Ce(17) [i] based on
(54, 54+25, large)-Net in Base 27 — Upper bound on s
There is no (54, 79, large)-net in base 27, because
- 23 times m-reduction [i] would yield (54, 56, large)-net in base 27, but