Best Known (8, 17, s)-Nets in Base 81
(8, 17, 1640)-Net over F81 — Constructive and digital
Digital (8, 17, 1640)-net over F81, using
- net defined by OOA [i] based on linear OOA(8117, 1640, F81, 9, 9) (dual of [(1640, 9), 14743, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(8117, 6561, F81, 9) (dual of [6561, 6544, 10]-code), using
- an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- OOA 4-folding and stacking with additional row [i] based on linear OA(8117, 6561, F81, 9) (dual of [6561, 6544, 10]-code), using
(8, 17, 2208)-Net over F81 — Digital
Digital (8, 17, 2208)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8117, 2208, F81, 2, 9) (dual of [(2208, 2), 4399, 10]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8117, 3281, F81, 2, 9) (dual of [(3281, 2), 6545, 10]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8117, 6562, F81, 9) (dual of [6562, 6545, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- OOA 2-folding [i] based on linear OA(8117, 6562, F81, 9) (dual of [6562, 6545, 10]-code), using
- discarding factors / shortening the dual code based on linear OOA(8117, 3281, F81, 2, 9) (dual of [(3281, 2), 6545, 10]-NRT-code), using
(8, 17, 1190974)-Net in Base 81 — Upper bound on s
There is no (8, 17, 1190975)-net in base 81, because
- 1 times m-reduction [i] would yield (8, 16, 1190975)-net in base 81, but
- the generalized Rao bound for nets shows that 81m ≥ 3 433693 670907 595685 330229 432001 > 8116 [i]