Best Known (48−24, 48, s)-Nets in Base 81
(48−24, 48, 547)-Net over F81 — Constructive and digital
Digital (24, 48, 547)-net over F81, using
- net defined by OOA [i] based on linear OOA(8148, 547, F81, 24, 24) (dual of [(547, 24), 13080, 25]-NRT-code), using
- OA 12-folding and stacking [i] based on linear OA(8148, 6564, F81, 24) (dual of [6564, 6516, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(8148, 6566, F81, 24) (dual of [6566, 6518, 25]-code), using
- construction X applied to Ce(23) ⊂ Ce(21) [i] based on
- linear OA(8147, 6561, F81, 24) (dual of [6561, 6514, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(8143, 6561, F81, 22) (dual of [6561, 6518, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(811, 5, F81, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(811, s, F81, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(23) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(8148, 6566, F81, 24) (dual of [6566, 6518, 25]-code), using
- OA 12-folding and stacking [i] based on linear OA(8148, 6564, F81, 24) (dual of [6564, 6516, 25]-code), using
(48−24, 48, 2034)-Net over F81 — Digital
Digital (24, 48, 2034)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8148, 2034, F81, 3, 24) (dual of [(2034, 3), 6054, 25]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8148, 2188, F81, 3, 24) (dual of [(2188, 3), 6516, 25]-NRT-code), using
- OOA 3-folding [i] based on linear OA(8148, 6564, F81, 24) (dual of [6564, 6516, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(8148, 6566, F81, 24) (dual of [6566, 6518, 25]-code), using
- construction X applied to Ce(23) ⊂ Ce(21) [i] based on
- linear OA(8147, 6561, F81, 24) (dual of [6561, 6514, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(8143, 6561, F81, 22) (dual of [6561, 6518, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(811, 5, F81, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(811, s, F81, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(23) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(8148, 6566, F81, 24) (dual of [6566, 6518, 25]-code), using
- OOA 3-folding [i] based on linear OA(8148, 6564, F81, 24) (dual of [6564, 6516, 25]-code), using
- discarding factors / shortening the dual code based on linear OOA(8148, 2188, F81, 3, 24) (dual of [(2188, 3), 6516, 25]-NRT-code), using
(48−24, 48, 2845841)-Net in Base 81 — Upper bound on s
There is no (24, 48, 2845842)-net in base 81, because
- the generalized Rao bound for nets shows that 81m ≥ 40 483927 766461 277035 548586 728704 422606 421121 729506 978546 401063 373404 806609 946048 421029 595521 > 8148 [i]