Best Known (54, 73, s)-Nets in Base 27
(54, 73, 59049)-Net over F27 — Constructive and digital
Digital (54, 73, 59049)-net over F27, using
- net defined by OOA [i] based on linear OOA(2773, 59049, F27, 19, 19) (dual of [(59049, 19), 1121858, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(2773, 531442, F27, 19) (dual of [531442, 531369, 20]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 531442 | 278−1, 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(2773, 531442, F27, 19) (dual of [531442, 531369, 20]-code), using
(54, 73, 318572)-Net over F27 — Digital
Digital (54, 73, 318572)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2773, 318572, F27, 19) (dual of [318572, 318499, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(2773, 531441, F27, 19) (dual of [531441, 531368, 20]-code), using
- an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 531440 = 274−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- discarding factors / shortening the dual code based on linear OA(2773, 531441, F27, 19) (dual of [531441, 531368, 20]-code), using
(54, 73, large)-Net in Base 27 — Upper bound on s
There is no (54, 73, large)-net in base 27, because
- 17 times m-reduction [i] would yield (54, 56, large)-net in base 27, but