Best Known (12, 12+13, s)-Nets in Base 81
(12, 12+13, 1093)-Net over F81 — Constructive and digital
Digital (12, 25, 1093)-net over F81, using
- net defined by OOA [i] based on linear OOA(8125, 1093, F81, 13, 13) (dual of [(1093, 13), 14184, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(8125, 6559, F81, 13) (dual of [6559, 6534, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(8125, 6561, F81, 13) (dual of [6561, 6536, 14]-code), using
- an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- discarding factors / shortening the dual code based on linear OA(8125, 6561, F81, 13) (dual of [6561, 6536, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(8125, 6559, F81, 13) (dual of [6559, 6534, 14]-code), using
(12, 12+13, 2187)-Net over F81 — Digital
Digital (12, 25, 2187)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8125, 2187, F81, 3, 13) (dual of [(2187, 3), 6536, 14]-NRT-code), using
- OOA 3-folding [i] based on linear OA(8125, 6561, F81, 13) (dual of [6561, 6536, 14]-code), using
- an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- OOA 3-folding [i] based on linear OA(8125, 6561, F81, 13) (dual of [6561, 6536, 14]-code), using
(12, 12+13, 1610910)-Net in Base 81 — Upper bound on s
There is no (12, 25, 1610911)-net in base 81, because
- 1 times m-reduction [i] would yield (12, 24, 1610911)-net in base 81, but
- the generalized Rao bound for nets shows that 81m ≥ 6362 691596 349290 310830 510117 846645 015850 613281 > 8124 [i]