Best Known (24, 24+25, s)-Nets in Base 81
(24, 24+25, 546)-Net over F81 — Constructive and digital
Digital (24, 49, 546)-net over F81, using
- net defined by OOA [i] based on linear OOA(8149, 546, F81, 25, 25) (dual of [(546, 25), 13601, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(8149, 6553, F81, 25) (dual of [6553, 6504, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(8149, 6561, F81, 25) (dual of [6561, 6512, 26]-code), using
- an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- discarding factors / shortening the dual code based on linear OA(8149, 6561, F81, 25) (dual of [6561, 6512, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(8149, 6553, F81, 25) (dual of [6553, 6504, 26]-code), using
(24, 24+25, 1640)-Net over F81 — Digital
Digital (24, 49, 1640)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8149, 1640, F81, 4, 25) (dual of [(1640, 4), 6511, 26]-NRT-code), using
- OOA 4-folding [i] based on linear OA(8149, 6560, F81, 25) (dual of [6560, 6511, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(8149, 6561, F81, 25) (dual of [6561, 6512, 26]-code), using
- an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- discarding factors / shortening the dual code based on linear OA(8149, 6561, F81, 25) (dual of [6561, 6512, 26]-code), using
- OOA 4-folding [i] based on linear OA(8149, 6560, F81, 25) (dual of [6560, 6511, 26]-code), using
(24, 24+25, 2845841)-Net in Base 81 — Upper bound on s
There is no (24, 49, 2845842)-net in base 81, because
- 1 times m-reduction [i] would yield (24, 48, 2845842)-net in base 81, but
- the generalized Rao bound for nets shows that 81m ≥ 40 483927 766461 277035 548586 728704 422606 421121 729506 978546 401063 373404 806609 946048 421029 595521 > 8148 [i]