Best Known (51, 51+18, s)-Nets in Base 27
(51, 51+18, 59049)-Net over F27 — Constructive and digital
Digital (51, 69, 59049)-net over F27, using
- net defined by OOA [i] based on linear OOA(2769, 59049, F27, 18, 18) (dual of [(59049, 18), 1062813, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(2769, 531441, F27, 18) (dual of [531441, 531372, 19]-code), using
- an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 531440 = 274−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- OA 9-folding and stacking [i] based on linear OA(2769, 531441, F27, 18) (dual of [531441, 531372, 19]-code), using
(51, 51+18, 316848)-Net over F27 — Digital
Digital (51, 69, 316848)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2769, 316848, F27, 18) (dual of [316848, 316779, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(2769, 531441, F27, 18) (dual of [531441, 531372, 19]-code), using
- an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 531440 = 274−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- discarding factors / shortening the dual code based on linear OA(2769, 531441, F27, 18) (dual of [531441, 531372, 19]-code), using
(51, 51+18, large)-Net in Base 27 — Upper bound on s
There is no (51, 69, large)-net in base 27, because
- 16 times m-reduction [i] would yield (51, 53, large)-net in base 27, but