Best Known (39−20, 39, s)-Nets in Base 81
(39−20, 39, 656)-Net over F81 — Constructive and digital
Digital (19, 39, 656)-net over F81, using
- net defined by OOA [i] based on linear OOA(8139, 656, F81, 20, 20) (dual of [(656, 20), 13081, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(8139, 6560, F81, 20) (dual of [6560, 6521, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(8139, 6561, F81, 20) (dual of [6561, 6522, 21]-code), using
- an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- discarding factors / shortening the dual code based on linear OA(8139, 6561, F81, 20) (dual of [6561, 6522, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(8139, 6560, F81, 20) (dual of [6560, 6521, 21]-code), using
(39−20, 39, 1666)-Net over F81 — Digital
Digital (19, 39, 1666)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8139, 1666, F81, 3, 20) (dual of [(1666, 3), 4959, 21]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8139, 2187, F81, 3, 20) (dual of [(2187, 3), 6522, 21]-NRT-code), using
- OOA 3-folding [i] based on linear OA(8139, 6561, F81, 20) (dual of [6561, 6522, 21]-code), using
- an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- OOA 3-folding [i] based on linear OA(8139, 6561, F81, 20) (dual of [6561, 6522, 21]-code), using
- discarding factors / shortening the dual code based on linear OOA(8139, 2187, F81, 3, 20) (dual of [(2187, 3), 6522, 21]-NRT-code), using
(39−20, 39, 1570278)-Net in Base 81 — Upper bound on s
There is no (19, 39, 1570279)-net in base 81, because
- the generalized Rao bound for nets shows that 81m ≥ 269 722921 880793 924284 848434 114928 464596 641216 015115 717736 735235 976223 903201 > 8139 [i]