Best Known (53, 53+19, s)-Nets in Base 27
(53, 53+19, 2369)-Net over F27 — Constructive and digital
Digital (53, 72, 2369)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (8, 17, 182)-net over F27, using
- net defined by OOA [i] based on linear OOA(2717, 182, F27, 9, 9) (dual of [(182, 9), 1621, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(2717, 729, F27, 9) (dual of [729, 712, 10]-code), using
- an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- OOA 4-folding and stacking with additional row [i] based on linear OA(2717, 729, F27, 9) (dual of [729, 712, 10]-code), using
- net defined by OOA [i] based on linear OOA(2717, 182, F27, 9, 9) (dual of [(182, 9), 1621, 10]-NRT-code), using
- digital (36, 55, 2187)-net over F27, using
- net defined by OOA [i] based on linear OOA(2755, 2187, F27, 19, 19) (dual of [(2187, 19), 41498, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on 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]
- OOA 9-folding and stacking with additional row [i] based on linear OA(2755, 19684, F27, 19) (dual of [19684, 19629, 20]-code), using
- net defined by OOA [i] based on linear OOA(2755, 2187, F27, 19, 19) (dual of [(2187, 19), 41498, 20]-NRT-code), using
- digital (8, 17, 182)-net over F27, using
(53, 53+19, 154396)-Net over F27 — Digital
Digital (53, 72, 154396)-net over F27, using
(53, 53+19, large)-Net in Base 27 — Upper bound on s
There is no (53, 72, large)-net in base 27, because
- 17 times m-reduction [i] would yield (53, 55, large)-net in base 27, but