Best Known (72, 72+14, s)-Nets in Base 27
(72, 72+14, 1204934)-Net over F27 — Constructive and digital
Digital (72, 86, 1204934)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (13, 20, 6563)-net over F27, using
- net defined by OOA [i] based on linear OOA(2720, 6563, F27, 7, 7) (dual of [(6563, 7), 45921, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(2720, 19690, F27, 7) (dual of [19690, 19670, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(2720, 19691, F27, 7) (dual of [19691, 19671, 8]-code), using
- construction X applied to C([0,3]) ⊂ C([0,2]) [i] based on
- linear OA(2719, 19684, F27, 7) (dual of [19684, 19665, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 276−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- 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(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([0,3]) ⊂ C([0,2]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2720, 19691, F27, 7) (dual of [19691, 19671, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(2720, 19690, F27, 7) (dual of [19690, 19670, 8]-code), using
- net defined by OOA [i] based on linear OOA(2720, 6563, F27, 7, 7) (dual of [(6563, 7), 45921, 8]-NRT-code), using
- digital (52, 66, 1198371)-net over F27, using
- net defined by OOA [i] based on linear OOA(2766, 1198371, F27, 14, 14) (dual of [(1198371, 14), 16777128, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(2766, 8388597, F27, 14) (dual of [8388597, 8388531, 15]-code), using
- discarding factors / shortening the dual code 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]
- discarding factors / shortening the dual code based on linear OA(2766, large, F27, 14) (dual of [large, large−66, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(2766, 8388597, F27, 14) (dual of [8388597, 8388531, 15]-code), using
- net defined by OOA [i] based on linear OOA(2766, 1198371, F27, 14, 14) (dual of [(1198371, 14), 16777128, 15]-NRT-code), using
- digital (13, 20, 6563)-net over F27, using
(72, 72+14, large)-Net over F27 — Digital
Digital (72, 86, large)-net over F27, using
- 5 times m-reduction [i] based on digital (72, 91, large)-net over F27, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(2791, large, F27, 19) (dual of [large, large−91, 20]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 14348908 | 2710−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(2791, large, F27, 19) (dual of [large, large−91, 20]-code), using
(72, 72+14, large)-Net in Base 27 — Upper bound on s
There is no (72, 86, large)-net in base 27, because
- 12 times m-reduction [i] would yield (72, 74, large)-net in base 27, but