Best Known (29, 59, s)-Nets in Base 81
(29, 59, 437)-Net over F81 — Constructive and digital
Digital (29, 59, 437)-net over F81, using
- net defined by OOA [i] based on linear OOA(8159, 437, F81, 30, 30) (dual of [(437, 30), 13051, 31]-NRT-code), using
- OA 15-folding and stacking [i] based on linear OA(8159, 6555, F81, 30) (dual of [6555, 6496, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(8159, 6561, F81, 30) (dual of [6561, 6502, 31]-code), using
- an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- discarding factors / shortening the dual code based on linear OA(8159, 6561, F81, 30) (dual of [6561, 6502, 31]-code), using
- OA 15-folding and stacking [i] based on linear OA(8159, 6555, F81, 30) (dual of [6555, 6496, 31]-code), using
(29, 59, 1689)-Net over F81 — Digital
Digital (29, 59, 1689)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8159, 1689, F81, 3, 30) (dual of [(1689, 3), 5008, 31]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8159, 2187, F81, 3, 30) (dual of [(2187, 3), 6502, 31]-NRT-code), using
- OOA 3-folding [i] based on linear OA(8159, 6561, F81, 30) (dual of [6561, 6502, 31]-code), using
- an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- OOA 3-folding [i] based on linear OA(8159, 6561, F81, 30) (dual of [6561, 6502, 31]-code), using
- discarding factors / shortening the dual code based on linear OOA(8159, 2187, F81, 3, 30) (dual of [(2187, 3), 6502, 31]-NRT-code), using
(29, 59, 2578595)-Net in Base 81 — Upper bound on s
There is no (29, 59, 2578596)-net in base 81, because
- the generalized Rao bound for nets shows that 81m ≥ 39867 400717 059294 883687 445834 203517 234674 548367 753622 323860 053284 706332 144794 993807 522889 203741 851911 475833 707201 > 8159 [i]