Best Known (81, 81+25, s)-Nets in Base 8
(81, 81+25, 2730)-Net over F8 — Constructive and digital
Digital (81, 106, 2730)-net over F8, using
- net defined by OOA [i] based on linear OOA(8106, 2730, F8, 25, 25) (dual of [(2730, 25), 68144, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(8106, 32761, F8, 25) (dual of [32761, 32655, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(8106, 32768, F8, 25) (dual of [32768, 32662, 26]-code), using
- an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- discarding factors / shortening the dual code based on linear OA(8106, 32768, F8, 25) (dual of [32768, 32662, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(8106, 32761, F8, 25) (dual of [32761, 32655, 26]-code), using
(81, 81+25, 17858)-Net over F8 — Digital
Digital (81, 106, 17858)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8106, 17858, F8, 25) (dual of [17858, 17752, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(8106, 32768, F8, 25) (dual of [32768, 32662, 26]-code), using
- an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- discarding factors / shortening the dual code based on linear OA(8106, 32768, F8, 25) (dual of [32768, 32662, 26]-code), using
(81, 81+25, large)-Net in Base 8 — Upper bound on s
There is no (81, 106, large)-net in base 8, because
- 23 times m-reduction [i] would yield (81, 83, large)-net in base 8, but