Best Known (27, 27+9, s)-Nets in Base 8
(27, 27+9, 8191)-Net over F8 — Constructive and digital
Digital (27, 36, 8191)-net over F8, using
- net defined by OOA [i] based on linear OOA(836, 8191, F8, 9, 9) (dual of [(8191, 9), 73683, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(836, 32765, F8, 9) (dual of [32765, 32729, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(836, 32768, F8, 9) (dual of [32768, 32732, 10]-code), using
- an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- discarding factors / shortening the dual code based on linear OA(836, 32768, F8, 9) (dual of [32768, 32732, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(836, 32765, F8, 9) (dual of [32765, 32729, 10]-code), using
(27, 27+9, 16384)-Net over F8 — Digital
Digital (27, 36, 16384)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(836, 16384, F8, 2, 9) (dual of [(16384, 2), 32732, 10]-NRT-code), using
- OOA 2-folding [i] based on linear OA(836, 32768, F8, 9) (dual of [32768, 32732, 10]-code), using
- an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- OOA 2-folding [i] based on linear OA(836, 32768, F8, 9) (dual of [32768, 32732, 10]-code), using
(27, 27+9, large)-Net in Base 8 — Upper bound on s
There is no (27, 36, large)-net in base 8, because
- 7 times m-reduction [i] would yield (27, 29, large)-net in base 8, but