Best Known (54, 65, s)-Nets in Base 27
(54, 65, 1687565)-Net over F27 — Constructive and digital
Digital (54, 65, 1687565)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (9, 14, 9845)-net over F27, using
- net defined by OOA [i] based on linear OOA(2714, 9845, F27, 5, 5) (dual of [(9845, 5), 49211, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(2714, 19691, F27, 5) (dual of [19691, 19677, 6]-code), using
- construction X applied to C([0,2]) ⊂ C([0,1]) [i] based on
- linear OA(2713, 19684, F27, 5) (dual of [19684, 19671, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 276−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- linear OA(277, 19684, F27, 3) (dual of [19684, 19677, 4]-code or 19684-cap in PG(6,27)), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 276−1, defining interval I = [0,1], and minimum distance d ≥ |{−1,0,1}|+1 = 4 (BCH-bound) [i]
- linear OA(271, 7, F27, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(271, s, F27, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,2]) ⊂ C([0,1]) [i] based on
- OOA 2-folding and stacking with additional row [i] based on linear OA(2714, 19691, F27, 5) (dual of [19691, 19677, 6]-code), using
- net defined by OOA [i] based on linear OOA(2714, 9845, F27, 5, 5) (dual of [(9845, 5), 49211, 6]-NRT-code), using
- digital (40, 51, 1677720)-net over F27, using
- net defined by OOA [i] based on linear OOA(2751, 1677720, F27, 11, 11) (dual of [(1677720, 11), 18454869, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(2751, 8388601, F27, 11) (dual of [8388601, 8388550, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(2751, large, F27, 11) (dual of [large, large−51, 12]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 14348908 | 2710−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(2751, large, F27, 11) (dual of [large, large−51, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(2751, 8388601, F27, 11) (dual of [8388601, 8388550, 12]-code), using
- net defined by OOA [i] based on linear OOA(2751, 1677720, F27, 11, 11) (dual of [(1677720, 11), 18454869, 12]-NRT-code), using
- digital (9, 14, 9845)-net over F27, using
(54, 65, large)-Net over F27 — Digital
Digital (54, 65, large)-net over F27, using
- t-expansion [i] based on digital (52, 65, large)-net over F27, using
- 1 times m-reduction [i] based on digital (52, 66, large)-net over F27, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(2766, large, F27, 14) (dual of [large, large−66, 15]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 14348906 = 275−1, defining interval I = [0,13], and designed minimum distance d ≥ |I|+1 = 15 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(2766, large, F27, 14) (dual of [large, large−66, 15]-code), using
- 1 times m-reduction [i] based on digital (52, 66, large)-net over F27, using
(54, 65, large)-Net in Base 27 — Upper bound on s
There is no (54, 65, large)-net in base 27, because
- 9 times m-reduction [i] would yield (54, 56, large)-net in base 27, but